V-shaped arrangements of turbines

ABSTRACT

A V-shaped arrangement of turbines adapted to orient in such a manner that crosswind enters the arrangement in a direction that is substantially parallel to the axis bisecting the arrangement is described. The V-shaped arrangement includes clockwise rotating turbines and counterclockwise rotating turbines.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a divisional of U.S. patent application Ser.No. 13/572,611 filed on Aug. 10, 2012, which claims priority to U.S.Provisional Application No. 61/523,164 filed on Aug. 12, 2011, both ofwhich are incorporated herein by reference in their entirety.

FIELD

The present disclosure relates to wind turbines. In particular, itrelates to v-shaped arrangements of turbines.

SUMMARY

According to a first aspect of the disclosure, a computer-implementedmethod for providing low-order potential flow elements for a verticalaxis turbine is described, the low-order potential flow elements usablefor configuring an array of vertical axis turbines. The method comprisesexecuting, using one or more computer systems, executable instructionsto perform at least the steps of receiving acquired flow data around aselected vertical axis turbine; assigning one or more variables, each toa set of potential flow elements, the one or more variables representingpotential flow around the selected vertical axis turbine; receiving andapplying a fitness function configured to evaluate a matching ofacquired flow data around the selected vertical axis turbine withpotential flow data given by the one or more variables representingpotential flow around the selected vertical axis turbine, according toone or more set thresholds, the set thresholds each being associated toa particular region around the vertical axis turbine; and applying analgorithm to the one or more variables, the algorithm configured tooutput values for each of the one or more variables which match theacquired flow data around the selected vertical axis turbine accordingto the fitness function, thus providing values for each of the one ormore potential flow elements representing potential flow around theselected vertical axis turbine.

According to a second aspect of the disclosure, a computer-implementedmethod of configuring an array of vertical axis turbines for an arraysite is described. The method comprises executing, using one or morecomputer systems, executable instructions to perform at least the stepsof receiving input data corresponding to potential flow around one ormore of a selected vertical axis turbine; receiving one or more inputvariables, the one or more input variables representing at least one ofa parameter of the array site, a constraint of the array site, and aparameter of the vertical axis turbine; receiving input concerning anumber of vertical axis turbines to be configured in the array;receiving input concerning one or more of a desired attribute of aconfigured array site; receiving input of a fitness function, thefitness function configured to rank configurations of array sites withinone or more constraints of the array site, according to one or more setthresholds, the thresholds being set according to a desired attribute ofa configured array site; and applying an algorithm to the one or morevariables, the algorithm configured to output data corresponding tolocations of vertical axis turbines which provide the one or more of adesired attribute of the array of vertical axis turbines.

According to third aspect of the disclosure, a further method forconfiguring an array of vertical axis turbines for a turbine array siteis described. The method comprises selecting a type of vertical axisturbine to be used in the array; providing an array of the selected typeof vertical axis turbines; providing one or more potential flow elementsrepresenting potential flow around the selected type of vertical axisturbine; selecting one or more input variables, the one or more inputvariables representing at least one of a parameter of the array site,and a constraint of the turbine array site, a parameter of the verticalaxis turbine; selecting a number of vertical axis turbines to be placedin the array; selecting a number of vertical axis turbines to be placedin the array site; designing a fitness function configured to rankconfigurations of a plurality of vertical axis turbine arrays within oneor more constraints of the array site, according to one or more setthresholds, the thresholds being set according to one or more of adesired attribute of a configured array site; inputting into analgorithm values for one or more potential flow elements representingpotential flow around the selected type of vertical axis turbine, theone or more input variables, and the number of vertical axis turbines tobe placed in the array site, wherein the algorithm is configured toprovide geometric data corresponding to locations of vertical axisturbines which provide a desired power output of the array of verticalaxis turbines; and configuring the array of the selected type ofvertical axis turbines according to the geometric data thus obtained.

According to a fourth aspect of the disclosure, a further method forconfiguring an array of vertical axis turbines for an array site isdescribed. The method comprises executing, using one or more computersystems, executable instructions to perform at least the steps of:receiving acquired flow data around a selected vertical axis turbineassigning one or more variables, each to a set of potential flowelements, the one or more variables representing potential flow aroundthe selected vertical axis turbine; receiving and applying a fitnessfunction configured to evaluate a matching of acquired flow data aroundthe selected vertical axis turbine with potential flow data given by theone or more variables representing potential flow around the selectedvertical axis turbine, according to one or more set thresholds, the setthresholds each being associated to a particular region around thevertical axis turbine; applying a first algorithm to the one or morevariables, the first algorithm configured to output values for each ofthe one or more variables which match the acquired flow data around theselected vertical axis turbine according to the fitness function, thusproviding values for each of the one or more potential flow elementsrepresenting potential flow around the selected vertical axis turbine;receiving the provided values for each of the one or more potential flowelements representing potential flow around the selected vertical axisturbine; receiving one or more further input variables, the one or morefurther input variables representing at least one of a parameter of thearray site, a constraint of the array site, and a parameter of thevertical axis turbine; receiving input concerning a number of verticalaxis turbines to be configured in the array; receiving input concerningone or more of a desired attribute of a configured array site; receivinginput of a fitness function, the fitness function configured to rankconfigurations of array sites within one or more constraints of thearray site, according to one or more set thresholds, the thresholdsbeing set according to a desired attribute of a configured array site;and applying an algorithm to the one or more variables, the algorithmconfigured to output data corresponding to locations of vertical axisturbines which provide the one or more of a desired attribute of thearray of vertical axis turbines.

According to a fifth aspect of the disclosure, a V-shaped arrangement ofturbines is described. The V-shaped arrangement of turbines comprises atleast three pairs of turbines, wherein: the V-shaped arrangement isadapted to be oriented such that a prevalent crosswind enters theopening of the V-shaped arrangement and such that the prevalentcrosswind direction is substantially parallel to the axis bisecting theV-shape; each turbine in a pair of turbines is proximate to the otherturbine in the pair and non-proximate to other turbines in the array;the vertex of the V-Shaped arrangement comprises a clockwise rotatingturbine from which a first arm of the V-shaped arrangement extends in afirst direction, and a counterclockwise rotating turbine, proximate tothe clockwise rotating turbine, from which a second arm of the V-shapedarrangement extends; turbines comprised in the first arm extending inthe first direction are clockwise rotating turbines; and turbinescomprised in the second arm extending in the second direction areclockwise rotating turbines.

The present disclosure provides various methods, which in someembodiments can be used in conjunction. A first method according toembodiments herein described can be used to model flow around a verticalaxis wind turbine (herein also referred to by the acronym “VAWT”) using,for example, potential flow elements including a vortex to capture therotation of the turbine, a dipole to capture a blockage effect of theturbine, a sink to capture the extracted energy from the wind by theturbine, and a source to capture the recovery of the flow due to inflowfrom around the turbines. These elements can be combined in differentnumbers or with different constraints in order to generate a model thatis fit to different data sets and that can optimize for computationalspeed or for accuracy. A second method provides configurations of a VAWTarray based on a desired attribute of the array, using a low-order modelwhen the array is subject to physical constraints. The physicalconstraints on the array can include, but are not limited to discretelocations at which the turbines can be placed, minimal or maximalspacing between turbines, a range and predominance of incoming winddirections, topography of the array site, and number and chirality ofturbines.

The details of one or more embodiments of the disclosure are set forthin the accompanying drawings and the description below. Other features,objects, and advantages will be apparent from the description anddrawings, and from the claims.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings, which are incorporated into and constitute apart of this specification, illustrate one or more embodiments of thepresent disclosure and, together with the description of exampleembodiments, serve to explain the principles and implementations of thedisclosure.

FIG. 1 shows a schematic of field array according to some embodiments.Open circles represent possible positions for turbines. Solid circlesrepresent counter-clockwise rotating turbine locations for experimentaldata acquisition; solid squares represent clockwise rotating turbinelocations for experimental data acquisition. Dashed lines indicate acenterline of the array. All dimensions are in English units. North istowards the top of the FIG. 1.

FIG. 2A shows a plot of experimental turbine velocity data. x indicatesthe horizontal (easterly) position, in turbine diameters. y indicatesthe vertical (northerly) position, in turbine diameters. Black circlesrepresent counter-clockwise rotating turbine locations, black squaresrepresented clockwise rotating turbine locations. The numbers next toeach turbine are the values of u_(∥), that is the velocity of theincoming wind, normalized to that of the first turbine in the array. Thegray squares represent the locations of the centerline datameasurements.

FIG. 2B shows a plot of experimental centerline velocity data.

FIGS. 3A-B shows plots from CFD convergence tests for a WINDSPIRE®turbine geometry. Time, forces, and grid cell size arenon-dimensionalized to the parameters of the system. In FIG. 3A, thesolid lines indicate the force on a blade in the x-direction as afunction of time. The value of h increases in the direction of the blackarrow with the values h=1.27* 10⁻², h=6.35*10⁻³, h=3.18*10⁻³,h=1.58*10⁻³, and h=7.94*10⁻⁴. The dashed lines indicate the time valuesat which the forces were evaluated for the plot on the right. In FIG.3B, solid lines indicate force on a blade in the x-direction as afunction of h. As indicated in FIG. 3A, (a) indicates values at timet=1, (b) indicates values at t=2, (c) indicates values at t=3, (d)indicates values at time t=4, and (e) indicates values at time t=5.

FIG. 4 shows a contour plot of vorticity from CFD simulation output fora single WINDSPIRE® turbine. The turbine is located at (0,0). The scaleof vorticity is such that light gray indicates a small magnitudevorticity and dark gray indicates a higher magnitude vorticity. Theinner rectangle indicates the boundary of the fine grid level. Thisframe of the flow was taken at non-dimensional time, t=10.

FIG. 5 shows a plot representing an analysis of cross-wind velocity intoturbines. Here the bold lines represent the turbine blades, the dotsmark the points at which the flow was analyzed (in actual tests, numberwas much larger), the solid arrows indicate the vector flow field ateach point, and the dashed arrows indicate the component of the vectorflow field “into” the turbine. The average of the magnitudes of thedashed arrows would define the flow velocity into the turbine, u_(∥).

FIG. 6 shows a plot representing averaged, thinned CFD velocity vectorfield for a single WINDSPIRE® turbine. The solid lines indicate theblades of the turbine, the light gray, boxed vector field indicates theregion of the flow designated as the wake region and the black vectorfield indicates the region of the flow designated as the surroundingregion.

FIGS. 7A-C show a comparison of velocity flow field for WINDSPIRE® CFDsimulation and fitted model. Black solid line indicates the x-componentof the simulated flow, black dotted line indicates the x-component ofthe modeled flow. Gray solid line indicates the y-component of thesimulated flow, gray dotted line indicates the y-component of themodeled flow. All velocities have been normalized to the incomingvelocity to the turbine in the simulated flow.

FIG. 7A shows a plot wherein each point on the plot represents theaverage of the respective flows over a cross-section of the entire fieldat the indicated x-position.

FIG. 7B shows a plot wherein each point on the plot represents theaverage of the respective flows over a cross-section of the bandimmediately in front of and behind the turbine at the indicatedx-position.

FIG. 7C shows a plot wherein each point on the plot represents the valueof the respective flow evaluated along the centerline at the indicatedx-position.

FIGS. 8A-B show plots wherein velocity is normalized to the valuemeasured experimentally immediately before the first turbine in thearray. FIG. 8A shows a plot representing an array centerline velocityprofile for a 30×30 array as computed by a potential flow modeloptimized for computational speed. The solid line is the velocity ascomputed from the model, the squares are the measurements taken on the3×3 array. FIG. 8B shows a plot representing an array centerlinevelocity profile for a 9×9 array as computed by potential flow modelsoptimized for computational speed or for physical accuracy. The solidline is the velocity as computed by the model optimized forcomputational speed, the dashed line is the velocity computed by themodel optimized for physical accuracy. The squares are the experimentalmeasurements from the 3×3 turbine array.

FIGS. 9A-B show plots representing empirical determination of scalinglaws from experimental data. Velocity is normalized to the valuemeasured experimentally immediately before the first turbine in thearray. In FIG. 9A, solid lines are centerline flow profiles array sizes(normalized to array diagonal length) increasing in direction of arrow:3×3, 4×4, 5×5, 7×7, 8×8, and 9×9. Dashed lines mark 1 array lengthupstream (a), 1 turbine diameter upstream (b), 1 turbine diameterdownstream (c), and 1 array length downstream (d). FIG. 9B showsevaluation of the centerline velocities at the points marked by thedashed lines in FIG. 9A for each array size and associated linear fits.

FIG. 10 shows a schematic of a predicted array for a single flowdirection using the method for providing a low order potential flowmodel and the method for providing a configuration of VAWT according thepresent disclosure. The solid circles represent a counter-clockwiserotating turbine, solid squares represent a clock-wise rotating turbine,the dots represent the possible locations at which the turbines canplaced, and the arrow indicates the prevalent wind direction for thisexample.

FIG. 11 shows a diagram showing performance of array optimized forsingle flow direction over a range of flow directions according tomethods herein described. Gray dots represent counter-clockwise rotatingturbines, while black dots represent clockwise rotating turbines. Thecircumferential axis is the angle in degrees from which the flow iscoming. The radial axis is the value of C_(p,array)′ attained by thearray when the flow is in the specified direction. The solid linerepresents the Savitzky-Golay filtered data using a polynomial of order2 and a window size of 11 data points.

FIGS. 12A-B show plots comparing centerline velocities for an 8×8turbine pair array as predicted by experimental and CFD derived models.The solid line indicates the experimentally derived model and the dashedline indicates the CFD derived model assuming each turbine is onlyaffected by the wake of the turbine directly upstream.

FIG. 12A shows a plot of the entirety of the data described above withreference to FIGS. 12A-B. FIG. 12B shows a plot of a close up of thedata described above with reference to FIGS. 12A-B to show details offit.

FIGS. 13A-B show plots representing empirical determination of scalinglaws from CFD data. Velocity is normalized to the value incoming to thefirst turbine in the array. In FIG. 13A, solid lines indicate centerlineflow profiles for (in direction of arrow) 3×3, 5×5, 7×7, 9×9 (black),15×15, 20×20, 30×30, 50×50, 75×75, and 100×100 arrays normalized toarray diagonal length. Dashed lines mark 1 array length upstream (a), 1turbine diameter upstream (b), 1 turbine diameter downstream (c), and 1array length downstream (d). FIG. 13B shows evaluation of the centerlinevelocities at the points marked by the dashed lines in FIG. 13A for eacharray size and associated linear fits.

FIG. 14 shows a plot comparing C_(p,array)′ values for square arrays ofcounter-rotating turbine pairs using a CFD model.

FIG. 15 shows diagram of the performance of array optimized according tomethods herein described for single flow direction over a range of flowdirections. The circumferential axis is the angle in degrees from whichthe flow is coming. The radial axis is the value of C_(p,array)′attained by the array when the flow is in the specified direction. Thesolid line indicates the values computed using turbine pair models, thedashed line indicates value computed using single turbine models. Theblack turbines are clockwise rotating and the gray turbines arecounter-clockwise rotating.

FIG. 16A shows a diagram showing C_(p,array)′ as a function of incomingwind direction for a counter-rotating turbine pair. Solid circlesrepresent counter-clockwise rotating turbines; solid squares representclockwise rotating turbines.

FIG. 16B shows a diagram showing C_(p,array)′ as a function of incomingwind direction for a co-rotating turbine pair of counter-clockwiserotating turbines. Solid circles represent counter-clockwise rotatingturbines.

FIG. 17 shows a flow diagram comprising steps for performing the methodof providing a low-order potential flow model of a vertical axis windturbine (VAWT) according to embodiments of the disclosure.

FIG. 18 shows a flow diagram comprising steps for performing the methodof configuring an array of vertical axis wind turbines (VAWTs) for anarray site according to embodiments of the disclosure.

FIG. 19 shows a schematic illustrating q_(sink). Arrow (20) indicates anincoming cross-wind approaching a wind turbine (22) at a first velocityand exiting at a second velocity (arrow (21)) which is lower than thefirst velocity as a result of energy taken out by the turbine (22).

FIGS. 20A-B shows schematic illustrations of a “blockage effect” or“dipole” (represented by μ) on two turbines (10). The schematicillustrations of FIGS. 20A-B show a view from above (aerial view) theturbines (10) looking down. FIG. 20A shows an example of a lower valueof μ wherein less blockage is observed as indicated by (11), compared toFIG. 20B which shows a higher value of μ and has a larger meandering offlow indicated by (12).

FIG. 21 shows a schematic illustrating recovery (q or q_(source)) ofwind flow as it passes through wind turbines (3). The arrows coming fromabove array (2) represent wind that comes from above the array ratherthan cross-winds (1). This can be referred to as turbulence and serves asource of energy separate from the main cross-winds and can allow for arecovery of wind speed between turbines (3). In FIG. 21, the X'sindicate geometric positions between wind turbines (3) and the plotbelow the turbines (3) indicates a variation in U (free-stream velocity)as a function of geometric position due to recovery (q_(source)). FIG.21 shows a general and approximate trend for change in U as a functionof geometric position in an array due to recovery; however, it is not aquantitative or exact representation.

FIGS. 22A-B show schematics of exemplary computer systems that can beused to carry out computer-implemented methods according to thedisclosure. FIG. 22A shows a computer system (30) which can be used tocarry out computer-implemented methods in some embodiments. FIG. 22Bshows a computer system (31) comprising a memory (32), a processor (33),a communications interface (34) and an interconnection arrangement (35)coupling the memory, processor and the communications interface. In FIG.22B, the memory can be encoded with instructions for executing one ormore steps of the methods of the disclosure.

FIGS. 23A-B show schematics of exemplary computer systems that can beused to carry out computer-implemented methods according to thedisclosure. FIG. 23A shows a single computer which can be used to carryout computer-implemented methods described herein. FIG. 23B shows aschematic of a computer system wherein the computer-implemented methodherein described is carried out on multiple computers in parallel.

DETAILED DESCRIPTION

Numerical modeling of vertical axis wind turbines (VAWTs) has typicallyfocused on a detailed modeling of individual wind turbines for use inoptimization of turbine design. Models that fall into this categoryinclude the actuator-disk model [8], the momentum model [10], thecascade model, and the vortex model [7]. Some more recently developedmodels are hybrids of these methods, for example, the momentum-vortexmodel [2]. One model has been developed which attempts to capture anoverall behavior of turbines using a minimal number of variables, thepotential flow model developed by Whittlesey et al. [15].

The potential flow model of Whittlesey et al. [15] consisted of auniform flow with a dipole and a point vortex superimposed to representone VAWT. The point vortex was used in order to account for the rotationcaused by the turbine and the dipole (aligned along the x-axis) was usedto account for the blockage due to the presence of the turbine in theflow.

Transcribed to the notation used in the present disclosure, theequations describing the flow elements in Whittlesey et al. [15] weregiven by:

$\begin{matrix}{F = {U_{\infty} - {\frac{\Gamma}{2\pi}{\log (z)}} + \frac{\mu}{2{\pi z}}}} & (1) \\{W = \frac{F}{z}} & (2) \\{u = {{(W)\hat{i}} - {i(W)\hat{j}}}} & (3)\end{matrix}$

where U_(∞) refers to the free-stream flow speed, Γ is the strength ofthe point vortex, μ is the strength of the dipole, î, ĵ are the unitvectors in the x,y directions, respectively, and z=x+iy (x and y arereal numbers.) In order to model an array of K turbines, the potentialflow elements were simply summed:

$\begin{matrix}{F = {{U_{\infty}z} + {\sum\limits_{k = 0}^{K}\; \left\lbrack {{1\frac{\Gamma}{2\pi}{\log \left( {z - z_{k}} \right)}} + {\mu \left( {z - z_{k}} \right)}^{- 1}} \right\rbrack}}} & (4)\end{matrix}$

where z_(k) corresponds to the location of the kth turbine. As for asingle turbine, equation 3 can be used to compute the velocity field. Toaccount for the velocity deficit in the wakes of the turbines,velocity-deficit curves [6] were applied:

u*(z)=(1−ξ_(ω)(z))u(z)  (5)

where u* is the resulting velocity vector after accounting for wakeeffects and ξ_(ω)(z) is a spatial function representing the deficit: anormal probability density function (PDF) in the lateral direction and abeta PDF in the downstream direction. Accounting for the wake in thismanner implies that u* is no longer a solution to the potential flowequations.

This model was least-squares matched to a small set of velocity vectorstaken in the vicinity of a functioning VAWT. Thus, for a given U_(∞), Γand μ can be found. The completed model was used to numerically evaluatean arrangement of turbines designed to mimic the swimming patterns offish schools that is closely spaced counter rotating pairs of VAWTs.Arrangements of turbines were compared by considering an arrayperformance coefficient and a power density parameter. The arrayperformance coefficient was defined to be:

$\begin{matrix}{C_{AP} = \frac{\overset{\_}{P_{array}}}{P_{iso}}} & (6)\end{matrix}$

where P_(iso) is the power output from a single isolated turbine andP_(array) is the average single turbine power output when the turbine isin an array (assuming that all turbines in the array are homogeneous).The power density parameter was defined as:

$\begin{matrix}{C_{PD} = {{\left( \frac{P_{array}}{A_{array}} \right)/\left( \frac{P_{iso}}{A_{iso}} \right)} = {{K \cdot C_{AP}}\frac{A_{iso}}{A_{array}}}}} & (7)\end{matrix}$

where P_(array), A_(array) are the total power and land area of thearray, respectively, A_(iso) is the land area of an isolated turbine,and K is the number of turbines.

It was found that for some spacings between turbines, the C_(AP) of thearrays comprising closely spaced counter-rotating pairs of turbines wasgreater than unity (i.e. the output from the array of K turbines wasmore than would be expected from K isolated turbines). A streamlineanalysis indicated that stream-tube contraction between pairs ofcounter-rotating turbines and the concomitant flow acceleration areprimarily responsible for the improvements in the expected power output.Further, a comparison of the power density of operative horizontal axiswind turbines (herein also referred to as the acronym “HAWT”) arrays toVAWT arrays indicate an order of magnitude better power output per unitland area. Dabiri [5] experimentally demonstrated the improvedperformance of pairs of VAWTs over isolated VAWTs and confirmed using asmall array of 6 turbines, that the power density performance of thearray was 6-9 times better than that of modern HAWT farms.

According to embodiments of the present disclosure, a method formodeling low-order potential flow of a vertical axis turbine isdescribed. In particular in some embodiments the vertical axis turbineis a vertical axis wind turbine (herein referred to as a “VAWT”). Inother embodiments, the vertical axis turbine is an underwater turbine.While various embodiments are described herein with reference to VAWTs,a skilled person will understand how to adapt embodiments described withreference to VAWTs to embodiments using vertical axis underwaterturbines, which are also within the scope of the present disclosure. Forexample, descriptions and examples herein with reference to a winddirection and/or source can be considered as being analogous, in someembodiments, to a direction and/or source of water current.

A low-order potential flow model of a VAWT according to some embodimentscan allow for potential flow around a VAWT to be represented by a smallset of numerical values, for example 3 to 4 numerical values.

In particular, embodiments of the present disclosure, allow for thepotential flow around a VAWT to be represented by a “blockage” effect(herein also referred to as a “dipole” and represented by μ), rotationalmotion (herein also referred to as a “vortex” and represented by F), anenergy sink (herein represented as “q” or “q_(sink)”) and an energysource (herein also referred to as “turbulence” and represented as “q”or “q_(source)”).

The “blockage” or (μ) effect can be described as being analogous towater flowing around a rock in a stream and is shown schematically inFIGS. 20A-B. For example, with a low value of μ, a small amount ofblockage is observed (11) in proximity to the turbine (10), while with ahigh value of μ, a big meandering of flow would be observed (12) inproximity to the turbine (10).

The rotational motion (Γ) can be described in line with the rotation ofthe VAWT. For example, if Γ is very small, there would be very littlerotational motion.

As shown schematically in FIG. 19, q, and in particular q_(sink), can bedescribed by considering that the velocity of wind flow into a wind farmis greater than the velocity that goes out based on the energy beingtaken out of the flow by the turbine. Therefore, an incoming windvelocity (20) can be greater than the velocity (21) of the wind afterpassing through one or more turbines (22). This loss in velocity of thewind can be represented by q_(sink).

In a wind farm, wind can be coming across a wind farm but can also becoming from above (herein also referred to as “turbulence”), and thiscan be described by q_(source). Therefore, q_(source) can provideadditional energy into a flow from turbulence and can lead to a recoveryof wind velocity through the wind farm. A contribution of energy to anarray of turbines by way of turbulence is shown schematically in FIG.21. In FIG. 21, U represents the velocity of a crosswind as it travelsthrough various positions (X) with respect to an array of turbines (3)and wind from above the array (turbulence) is indicated with curvyarrows (2). FIG. 21 schematically shows a crosswind (1) velocitydecreasing after passing through a first turbine and then increasingbefore reaching the next turbine, which is referred to as “recovery” andcan be attributed to the turbulence.

According to embodiments herein described, q_(source) is a parameter,which can become more relevant in embodiments in which an arrangement ofturbines rather than a single turbine is being considered.

According to some embodiments herein described, a potential flow modelaround a selected VAWT can be provided by first determining whichnumbers corresponding to μ, Γ, q_(sink), and q_(source), can be attachedto a given flow and then using an optimization, for example a localoptimization (e.g. a genetic algorithm) to find which values for theseparameters will give the best match either simulated or experimentaldata representing wind flow around the selected VAWT.

According to further embodiments of the disclosure, the potential flowmodel for a given VAWT can be used to provide arrays based on a desiredattribute of the array, for example, maximizing power output per unitland area of the array, minimizing environmental impact, or minimizingacoustic, radar, or visual impacts. In these embodiments, a localoptimization can be used (e.g. a genetic algorithm) to predict ageometrical arrangement of turbines, which optimize (at least locally)the power output of the array.

The term “potential flow” as used herein refers to a possibly nonsteady,multidimensional flow of a fluid having a substantially irrotationalvelocity field and being substantially frictionless. Potential flow canbe used to describe a velocity field as a gradient of a scalar function(e.g. velocity potential). For example, potential flow can be used todescribe an outer flow field for airfoils such as a wing or a blade(e.g. of a turbine), water waves, electroosmotic flow, and ground waterflow. One feature of potential flow is that the combined effect of aplurality of flows due to a corresponding plurality of objects in theflow (e.g. VAWTs) can be calculated by a mathematical addition of theindependent flows created by each individual object. Therefore, inembodiments, where flow is simulated, simulating potential flow canreduce a computational cost of simulating the flow around multipleobjects (e.g. VAWTs).

The term “potential flow element” as used herein refers to a parameterused to represent potential flow around an object. For example, a dipoleμ, can be added to a potential flow expression in order to describe avelocity field moving around an object and the dipole term can bereferred to as a “potential flow element”. Potential flow elements canbe used, for example, in a generation of a flow field and can includeone or more a vortex Γ, a dipole μ, a source q_(source), and a sinkC_(sink).

The term “fitness function” as used herein is refers to a mathematicalexpression, which selects a best element or best set of elements withrespect to some criteria from some set of available alternativeelements. For example, a fitness function can be used to determine howclose one or more solutions are to achieving a set goal, for example, byassigning a score to each of the one or more solutions, the score beingindicative of how close the one or more solutions are to achieving a setgoal. The set goal can be, for example, a matching with a set value(e.g. a numerical value). For example, the solutions can be ranked frombest to worst based on their fitness values as calculated by the fitnessfunction.

The term “genetic algorithm” as used herein refers to a search heuristicwhich is designed to mimic the process of evolution and typicallycomprises use of a genetic representation of a solution domain and afitness function to evaluate the solution domain. For example, a fitnessfunction can be defined over a genetic representation and can be set tomeasure the quality of a represented solution. In initial steps of agenetic algorithm, a number of individual solutions are can be randomlygenerated to form an initial population or can be “seeded” in areaswhere optimal solutions are likely to be found or have been previouslyfound through results of genetic algorithm, experimental results,simulated results, a physical limitation, a calculation, or an educatedguess.

The term “free stream velocity” as used herein refers to the velocity ofundisturbed fluid flow (either real or modeled). For example, if a gustof wind moves towards a turbine array, before it reaches the array, itcan be considered to be moving at “free stream velocity”. As anotherexample, wind movement to either side of a turbine array that isunaffected by the array can also be considered to be moving at “freestream velocity”.

The term “low-order” as used herein can be used to describe acomputational model, which represents a physical system using a minimalor relatively low number of variables, for example compared to asimulated model. For example, a low-order computational model can allowcomputations involving the model to be more rapidly evaluated comparedto a higher order model such as a simulated model. A low-order modelgenerally describes the physical system in less detail than a higherorder model, focusing instead on generating a more global, bird's-eye,view of the physical system.

The term “optimization” as used herein refers an optimization which isdirected to a desired effect, for example, power output of a turbinearray or power output within one or more constraints of a system forwhich the optimization is being performed (e.g. configurations ofturbines). Such constraints can vary and can be defined by a user andtherefore, any given optimization can be considered as an optimizationaccording to user input. Therefore an “optimization” as herein describedis a relative term, which can vary depending on, for example,constraints of the system. Further, the optimizations herein describedrefer to local optimizations.

The term “local optimization” as used herein refers to an optimalsolution to a problem or equation with respect to a neighboring set ofsolutions to the same problem or equation.

According to the present disclosure, a method is provided for modelingpotential flow of turbines, which can in some embodiments becomputationally simple. In some embodiments, the method can be used in afurther method for providing arrangements of a large number of turbineswhich can be computed relatively quickly and allow for examination ofmultiple configurations of turbines as well as different flow directionsinto the array in order to find a desired arrangement of the turbinesbased on a desired attribute of the turbine array.

The method for providing a low-order potential flow model of a verticalaxis wind turbine comprises adopting one or more of a selected VAWT unitto serve as a model. In some embodiments, a single VAWT can serve as themodel. In other embodiments, a pair of VAWTs can serve as the model. Inthese embodiments, the pair of VAWTs can be co-rotating VAWTs orcounter-rotating VAWTs. According to further embodiments, variouscombinations of a single turbine, a pair of co-rotating turbines, and apair of counter-rotating turbines can be used together.

The method further comprises assigning a variable to one or morepotential flow elements to represent potential flow around the selectedVAWT. Potential flow elements can include, for example, rotational (Γ),a blockage effect (μ), energy taken in by the VAWT (q or q_(sink)), andturbulence (q or q_(source)). The number of potential flow elements tobe used in representing potential flow around a selected VAWT can dependon whether it is desired to obtain values for the potential flowelements with a higher computational speed or whether it is desired toobtain set of values which can more accurately represent the turbine.Further, in connection with using these values to obtain a configurationof turbines in an array according to embodiments herein described.Eliminating a potential flow element can decrease the accuracy of thealgorithm and increase computational speed. Adding potential flowelements can increase the accuracy of the algorithm and decreasecomputational speed. Therefore, the number and type of potential flowelements can depend on a desired balance between speed and accuracy ofthe computation.

In some embodiments, a potential flow element can be included orexcluded based on an expected contribution of the potential flowelement. For example, if potential flow elements which are expected tohave a value close to zero can be excluded to decrease computation timewith substantially a same accuracy as can be obtained by including thepotential flow element. For example, if an array site is expected tohave little or no turbulence, then q_(source) can be excluded todecrease computational time. As a further example, if an array site isexpected to a high level of turbulence, q_(source) can be included toobtain higher accuracy, and in some embodiments, more than oneq_(source) can be included.

In potential flow models according to some embodiments herein described,the complex potentials (F) and complex velocities (W=dF/dz) of eachelement can be estimated with the following equations.

Uniform Flow:

F=U _(∞) z

W=U _(∞)  (8)

Vortex:

$\begin{matrix}{{F = {{- }\frac{\Gamma}{2\pi}{\ln \left( {z - z_{0}} \right)}}}{W = {{- }\frac{\Gamma}{2{\pi \left( {z - z_{0}} \right)}}}}} & (9)\end{matrix}$

Dipole:

$\begin{matrix}{{F = {\frac{\mu}{2{\pi \left( {z - z_{0}} \right)}}}}{W = {{- }\frac{\mu}{2{\pi \left( {z - z_{0}} \right)}^{2}}}}} & (10)\end{matrix}$

Source/Sink:

$\begin{matrix}{{F = {\frac{q}{2\pi}{\ln \left( {z - z_{0}} \right)}}}{W = \frac{q}{2{\pi \left( {z - z_{0}} \right)}}}} & (11)\end{matrix}$

In this notation, Γ represents the strength of a vortex, μ representsthe strength of a dipole, q represents the strength of a source ifpositive and the strength of a sink if negative, z is the point in thecomplex plane at which the flow is being evaluated (z=x+iy, where x,yε

), and z₀ is the location in the complex plane of the element beingevaluated. In embodiments where multiple elements are present, thecorresponding F and W can be summed. Once W has been computed, thevelocity flow field u can be represented by:

u=

(W)î−iℑ(W)ĵ  (12)

Here î and ĵ represent the unit vectors in the x and y directions,respectively, and

, ℑ are the real and imaginary component operators, respectively.

The method herein described further comprises designing a fitnessfunction adapted to allow a matching of acquired flow data around theselected VAWT unit with potential flow data given by the one or morevariables representing potential flow around the selected VAWT, withinone or more set thresholds, the set thresholds being associated to aparticular region around the VAWT.

The method herein described further comprises applying an algorithm tothe one or more variables, the algorithm configured to output numericalvalues for each of the one or more variables which match the acquiredflow data around the selected VAWT unit according to the designedfitness function, thus providing numerical values for each of the one ormore potential flow elements representing potential flow around theselected VAWT.

By way of example and not of limitation, a method for modeling potentialflow of a VAWT cross-section according to some embodiments is nowdescribed.

In order to generate a model, a number of each type of element, aposition of each element, and a strength of each element in a potentialflow can be optimized such that a resulting flow field designated hereinas “u_(model)”, matches as closely as possible to original or “acquired”data.

In some embodiments, the acquired data is experimental data, hereindesignated “u_(exp)”. In other embodiments the acquired data issimulated data, herein designated “u_(sim)”.

For example, in order to acquire experimental data, a particular type ofVAWT and number of VAWTs in array can be arranged on a desired arraysite to form a turbine array. The particular type of turbine selectedcan vary in height, diameter, chirality, and cut in wind speeds, forexample. The turbines can be arranged, for example, according toconstraints of the array site, such as size of the array site, topologyof the array site, and available positions in which the turbines canmounted and/or based on constraints of the VAWT, such as diameter of theturbine, and number of turbines.

In particular, constraints on a system can be input into a computationas bounds on a variable space over which the optimization algorithm isperformed. For example, if it was desired to contain the array within aplot of land of known dimension, but the turbines can be anywhere withinthat plot of land, the algorithm can take the positions of the turbinesas variables and the possible range of positions (e.g. based on the sizeof the available plot of land) can be set as limits on which values thevariable can attain.

Similarly, constraints such as size of the array site, availablepositions in which the turbines can be mounted, number of turbines,chirality of turbines, and diameter of turbines, which can be consideredas limits in the physical system (e.g. having little or no flexibility)can be input into the optimization algorithm as bounds on respectivevariables.

Constraints on a physical system, which are considered to be flexible,can be put into the optimization as modifiers of the fitness function.For example, if there is a hill near the center of the land plotavailable, and turbines can be placed behind the hill with respect to acertain wind direction, a fitness function can be designed such that ifa turbine position was behind the hill, the evaluated incoming windvelocity would be modified by some set amount to account for the windblockage from the hill.

In some embodiments, a three-dimensional nature of the flow (e.g. due totopography) can be accounted for, for example, by stacking a pluralityof two dimensional models, one on top of another and passing informationfrom one layer to another.

According to embodiments herein described, turbines can arranged as aseries of single turbines, as pairs of turbines (either co-rotating orcounter-rotating), or in another group of turbines that are spacedclosely enough that non-linear fluid flow interactions can besubstantial. Non-linear fluid flow interactions can be considered to besubstantial if the flow around each turbine in a group of turbines, whenadded together, does not provide a desired level of accuracy (e.g. a“non-linear interaction”) in physically representing the flow around thegroup of turbines. This can be due to influences between proximateturbines.

For example, by analogy, a first rock in a stream can provide a firstflow pattern and second rock far downstream from the first rock canprovide a second flow pattern which can be similar to the first patternand thus both flow patterns can be represented by a same model. If thefirst rock and the second rock are closer together (e.g. one rockimmediately behind or in front of another), the flow around each rockcan be altered. For example, at least part of the flow directly behindthe first rock will not be seen by the second rock; therefore, in thisexample, adding the models for each individual rock can considered asnot accurately representing the two rocks. Therefore, the farther apartthe two rocks, the more accurately the two rocks can be represented by amodel which adds two individual models.

In some embodiments, potential flow elements can be provided for asingle turbine or a group of proximate turbines (e.g. to account fornon-linear interactions between proximate turbines). In some embodimentsthe groups of proximate turbines are a pair of co-rotating turbines, apair of counter-rotating turbines, and/or another group of turbinesrotating in various respective directions. According to some embodiment,an increase in the number of turbines can lead to an increase incomputational time. Therefore, in embodiments where shortercomputational times are desired, a lesser number of turbines in thegroup can be used.

FIG. 1 shows a schematic of an exemplary turbine array, which can beused for obtaining experimental data. In particular, FIG. 1 shows anarray with eighteen turbines, nine counter-clockwise rotating and nineclockwise rotating. As would be understood by a skilled person, variousother arrangements can be used.

In order to obtain experimental data from a turbine array, measurementscan be obtained for parameters of a turbine including but not limited toelectrical power generated by each turbine, turbine rotational speed,turbine aerodynamic torque, turbine acoustic signature, and turbineradar cross-section. Each of these parameters, either alone or incombination, can be used to match acquired experimental data withpotential flow data given by one or more variables representingpotential flow around the turbine.

According to some embodiments herein described, a selection of one ormore parameters of a turbine for which to obtain measurements can bebased on which attributes of a configuration of turbines is desired tobe optimized and/or compared with other configurations of turbines. Forexample, power output measurements can be used to provide a power-outputoptimized configuration of turbines and/or to compare configurations ofturbines based on the power output of each configuration, and anacoustic output measurement can be used, for example, to optimize for aquieter configuration of turbines and/or to compare configurations ofturbines based on acoustic output of each configuration.

According to a further embodiment, the measurements of parameters of theturbine can be used to design a more realistic fitness function.

In a further embodiment, the measurements can be used to provide adatabase of potential flow elements in connection with various physicalaspects of a turbine (e.g. rotational speed, cross section, bladenumber, blade diameter, and/or height). For example, the potential flowelements in connection with various physical aspects of a turbine can beprovided according to methods herein described. In these embodiments,physical aspects of a selected turbine to be placed in an array can becompared to the database and potential flow elements which can be usedto represent flow around the selected turbine can be directly matched toturbine based on its physical aspects.

In some embodiments, the electrical power generated by each turbine canbe monitored in real-time and recorded, for example using power meters.In these embodiments, power meters can be connected to a central datalogger. Various other methods for obtaining experimental data from aturbine array are identifiable by a skilled person.

In some embodiments a meteorological tower can be used to collectexperimental data from a turbine array. The data obtained in this waycan be used to estimate the wind speeds within the array, for example,along the centerline of the array.

In these embodiments, a single meteorological tower can be used or aplurality of meteorological towers can be used. For example a pluralityof meteorological towers can be placed at a plurality of positionsthroughout the array to obtain measurements. As another example, asingle meteorological tower can be moved from a first position to aseries of subsequent positions after obtaining measurements at the firstposition and each subsequent position and can continue to be moved inthis way until measurements at any desired locations is obtained. Datafrom one or more meteorological towers can also be recorded using adatalogger.

Experimental data can be obtained over a period of time ranging fromminutes to years, depending on the inherent variability of the wind at aparticular site as would be understood by a skilled person. For example,the more variable the wind is at a particular site, the longer it cantake to obtain which can be used to represent the potential flowelements with a desired level of accuracy. An exemplary method forobtaining experimental data is shown in Example 1.

According to some embodiments, in relating the power output of a turbine(P) obtained experimentally to flow conditions, it can be estimated thatP∝u_(∥) ³ where u_(∥) is the flow directed into the turbine (i.e.parallel to the free-stream direction).

As previously mentioned, in some embodiments the acquired data issimulated data. In these embodiments, simulated data can be obtainedthrough various simulations, including but not limited to CFDsimulations, Monte-Carlo simulations, molecular dynamics simulations,and equation-free simulations. These simulations, and other simulationsidentifiable by a skilled person, can have varying degrees of speed andaccuracy. Therefore, a selection of one of these simulations to be usedaccording to embodiments of the present disclosure, can be based onwhether it is desired to have a higher speed simulation or whether it isdesired to have a simulation which can provide a higher accuracy.

In these embodiments, a flow field around the VAWTs can be obtained bytwo-dimensional simulations of at least one of a single VAWTcross-section, a pair of co-rotating turbines, and a pair ofcounter-rotating turbines. For a more complete flow field around theVAWT, simulations of the single VAWT cross-section, the pair ofco-rotating turbines, and the pair of counter-rotating turbines can allthree be obtained. For example, in embodiments where accuracy is moreimportant, all three simulations can be obtained. In embodiments where ashort time is important, simulation of a single VAWT cross-section canbe used.

In some embodiments, for simulation of one or more VAWT cross-sections,an arbitrary geometry and convergence conditions can be defined.

For example, in embodiments where CFD simulations are used animmersed-boundary Navier-Stokes solver can be used to compute atwo-dimensional flow around arbitrarily defined geometries. In someembodiments, the geometry can be made to resemble cross-sections of theturbines used for experimental data such that two related sets ofacquired data are obtained.

A grid size and time step can be selected to allow convergence of theflow. A grid size and time step can be selected, for example, such thatgiven basic physical properties of a simulated fluid and simulated flowconditions, a same flow solution is found for smaller values of the gridsize and time step. This example can be considered to be a convergedsolution and can indicate that the simulated flow is being adequatelytracked from one grid cell and time step to the next grid cell and timestep.

In some embodiments, the Courant-Friedrichs-Lewy (CFL) condition can beused to determine the numeric stability of the simulation and is definedto be

$\begin{matrix}{{CCFL} = \frac{U*\Delta \; t}{h}} & (13)\end{matrix}$

where U is represents a non-dimensional velocity (maximum velocity ofthe turbine/incoming free stream velocity), Δt represents anon-dimensional time step (dimensional time x incoming free streamvelocity/diameter of the turbines), and h represents a non-dimensionalgrid cell size (dimensional size/turbine diameter). For the simulationto be numerically stable, for example such that errors in the simulatedsystem do not grow exponentially, it is can be set that CFL≦0.7. In someembodiments, absolute values of Δt and h can be varied so that theCFL≦0.7 ratio will hold.

In an exemplary embodiment, the simulations are run at CFL=0.25 varyingabsolute values of Δt and h (See e.g. Example 2).

In some embodiments, maximum dimensional velocity of a system can beconservatively taken to be the free-stream velocity plus the tipvelocity of the rotating turbines plus a margin of error of around 0.5m/s. In order to analyze convergence, the forces on one of the blades ofthe turbine can be examined over time.

In order to limit computational time, simulations can be run with morethan one grid level. For example two or more grid levels can be usedwherein a fine grid is used close to the turbine and a coarse grid isused over an entire simulated domain.

A fine grid can be used over relatively short distances, for example, infront of and to either side of the turbine and relatively long distancesdownstream of the turbine as these areas can be most affected by theturbine (See e.g. FIG. 6). For example, in some embodiments a fine gridcan be used over approximately 0.5 D, D representing a turbine diameter,in front of and to either side of the turbine and approximately 5 Ddownstream from the turbine.

A coarse grid can be substantially larger, covering areas outside thosedescribed above with respect to the fine grid. For example, the coarsegrid can be just over approximately 3.5 D upstream, approximately 1.5 Dto either side of the turbine and approximately 8 D downstream. Othervariations are identifiable by a skilled person and can depend onfactors such velocity of cross winds and/or computational factors suchas time and/or accuracy of the simulations. For example, if the flow issimulated for higher velocity cross winds, the fine grid can be extendedfurther downstream than for a flow which is simulated for a lowervelocity cross wind.

A matching of experimental data to provide a model of potential flow isnow described by way of example.

In some embodiments a model can be evaluated and compared to othermodels by a root-mean-square difference between the experimental datapoints and corresponding model data points. In these embodiments, thefitness function of the genetic algorithm can be set to minimize thisRMS difference.

In some embodiments, centerline data velocities are matched. Thecenterline velocity can be used can be used to describe the velocityprofile of a VAWT array. In particular, the velocity profile canindicate how velocity of a cross wind changes as a function of positionthroughout the VAWT array. For example, the velocity profile can showhow the velocities drop as the array progresses (which can be considereda significant characteristic of the array. Therefore, the centerlinevelocity can be used as a method to evaluate and compare a plurality ofVAWT arrays having different configurations. Experimentally acquiredcenterline velocities can be obtained, for example, from meteorologicaltower measurements as herein described.

In some embodiments, in matching experimental turbine velocity data,which can be taken to be the cube root of the measured power data, aline of data points can be taken perpendicular to the free-stream flowdirection and spanning from one edge of a turbine to another. Thecomponent of potential flow perpendicular to this line (parallel to thefree-stream direction) can then be determined at each of the test pointsand averaged in order to determine the flow velocity into each turbineas a single number, u_(∥), see for example, FIG. 5.

Several embodiments of the disclosure allow a reduction of a number ofvariables for modeling a turbine. For example, in some embodiments,models estimate that there is a vortex and a dipole at the center ofeach turbine and thus a total number of vortices and dipoles isequivalent to the number of turbines included in the array.

The sinks can either be “free-floating”, for example, a positionoptimized within a small box centered on a turbine or “fixed”, forexample, a position either predetermined or determined by modeled flow.A selection of the sinks to be either free-floating or fixed can bebased on whether it is desired to have a model with greater physicalaccuracy, in which case free-floating sinks can be used, or a modelwhich can be calculated with higher computational speed and can thus beprovided in less time, in which case the sink can be fixed.

The sources can either be modeled as fixed, free-floating (e.g. within asame range as a sink) or as a source field over an entire arraycomprising sources of equal strength placed in a regular grid. Hereagain, a selection of the sources to be free-floating, fixed, or as asource field over an entire array can be based on whether it is desiredto have a model with greater physical accuracy or a model, which can beprovided in less time. Once again, the free-floating sources can providea model with higher accuracy while using either a fixed source or asource filed over an entire array, can provide a model in less time.Between a fixed source and source over an entire array, the fixed sourcecan provide a model with higher accuracy while the source over an entirearray can provide a model in less time.

As already described herein, a vortex element can be used to model,physically, a rotational component introduced to the flow by a rotatingturbine, a dipole element can be used to account for “blockage” effectsof the turbine, a sink can be used to account for an extraction ofenergy from the wind by the turbine, and a source can be used tosimulate flow recovery due to entrainment of free stream flow in wake.

In some embodiments, a source field can be used to simulate inflow ofair from above the turbine array [4] rather than a single source. Inthese embodiments, a uniform field can be estimated.

In some embodiments, a selected location of the sink is approximately 1D-4 D downstream, and more particularly in some embodiments, the sink islocated 2 D downstream of each turbine center [1]. The range of 1 D-4 Dfor placement of a sink can be a suitable placement for a sink accordingto some embodiments, because there can be a complex wake regionextending over approximately 2-4 diameters downstream of a wind turbine.The wake in this region can be due to a relaxation of axial and radialpressure gradients due at least in part to an extraction of energy fromturbine, which can cause the centerline velocity to drop. A minimumcenterline velocity can be reached between 1 D-2 D downstream, beyondwhich the velocity begins to recover [1].

In some embodiments, the model according to embodiments herein describedcan be used to predict air flow about larger turbine arrays. In theseembodiments, the algorithm as described herein can be modified toaccount for linear decreases in a centerline flow velocity to valuesbelow zero for predicted arrays having larger numbers of turbines (Seee.g. FIG. 8).

In these embodiments, an iteratively defined strength model can be usedwherein strengths of the vortices and sinks for each turbine can beestimated to be proportional to the incoming velocity raised to anumerical power; that is, if the incoming flow velocity to turbine j isgiven by v_(j), then Γ₁∝av_(j) ^(x); and q_(j)∝bv_(j) ^(y) where a,b aresome constants of proportionality and x,y are numerical exponents.Similarly the strength of the dipole for each turbine can be estimatedto be proportional to the incoming velocity: μ_(j)∝cv_(j) ^(z), where cis some constant of proportionality and z is a numerical exponent. Inthis example, using these proportionalities with x, y=3 and z=2, thefour variables of the model are then three constants of proportionalityand the strength of the sources in a field array.

In these embodiments, the algorithm can use iteration betweendetermining the strengths of the elements and the flow field velocities,which can be more computationally intensive than a homogenous strengthmodel. Using iteration as described here can result in more accuratepredictions for larger arrays.

A matching of simulated data to provide a model of potential flow is nowdescribed by way of example.

A first step in matching simulated data can be to average the simulateddata over a simulated time span in order to obtain a time-averaged flowpattern. In some embodiments, a time-varying potential flow model can beused in place of a time-averaged flow pattern.

Depending on resolution of the simulated data, a further step in thematching of simulated data can comprise down sampling the simulated datato a lower spatial resolution to simplify computations. An exemplarysimulated flow field for a single WINDSPIRE® turbine is shown in FIG. 6.

In some embodiments, a wake region and surrounding region of the flowfield can be modeled with separate potential flow models. For example,if a single potential flow model does not provide results having adesired level of accuracy (e.g. as evaluated by an RMS between thesimulated field and the modeled field), then the wake region andsurrounding region can modeled with separate potential flow models.

In embodiments using simulated turbine data, the number of variables canbe reduced by assuming one vortex and one dipole at the center of eachturbine.

The fitness functions for the surrounding region and the wake region canbe designed differently in order to emphasize different characteristicsof the flow. For example, differences between the simulated and modeledflow velocities downstream, where another turbine might be placed, canbe considered to be more important in terms of a matching than otherregions of the flow. As another example, in some embodiments, it wouldbe less important to have the simulated data match the modeled flowvelocities in direct proximity to a turbine where another turbine wouldnot be placed. Other features for modifying a fitness function accordingto various regions within an array would be understood and identifiableby a skilled person upon reading the present disclosure.

In some embodiments a model can be evaluated and compared to othermodels by taking the RMS differences (for each flow direction) betweenan average at cross sectional slices of the simulated field and themodeled field.

According to some embodiments, models based on simulated data, using avortex and dipole at the center of each turbine and a plurality offree-floating sources/sinks rather than a source field and fixed sinkprovides a better match between the simulated and the modeled flow field(See e.g. Example 6).

In some embodiments, the variables for which optimization is performedcan be fixed prior to model parameter optimization. Even in some ofthese embodiments, the number of variables can still be significant anda grid search algorithm can become computationally unfeasible as thenumber of variables increases. Therefore, in some embodiments a geneticalgorithm search can be used locally optimize model parameters.

Other local optimization algorithms can also be used without departingfrom the scope of the present disclosure. The exact structure of theoptimization algorithm is not necessarily relevant to the overalloperation of the presented algorithms. Other algorithms suitable foroptimization of model parameters are identifiable by a skilled person,such as grid-search algorithms, gradient-based algorithms, the surrogatemanagement framework, and simulated annealing, for example. Thesealgorithms can have varying degrees of speed and accuracy. Therefore, aselection of one of these algorithms to be used according to embodimentsof the present disclosure can be based on whether it is desired to havea higher speed algorithm or whether it is desired to have an algorithmwhich can provide a higher accuracy.

Steps which can be performed in a genetic algorithm are now described,by way of example and not of limitation, and the following descriptionis adapted from MATLAB® documentation [13].

A genetic algorithm can begin by creating a random initial “population”.An “individual” as used herein, refers to a set of numbers, one numberfor each variable of a problem. A population then refers to a group ofindividuals. The algorithm can then create a sequence of newpopulations.

At each step, the algorithm can use the individuals in a currentgeneration to create a subsequent population. To create a newpopulation, the algorithm can perform, for example, the following steps:scoring each member of the current population by computing its fitnessvalue (using a fitness function written by a user specially adapted forthe optimization); scaling raw fitness scores to convert them into amore usable range of values; selecting members, called parents, based ontheir fitness; choosing “elite” individuals in the current populationthat have lower fitness; passing elite individuals on to the nextpopulation; and producing children from the parents.

Children can be produced, for example, either by making random changesto a single parent, referred to as “mutation” or by combining vectorentries of a pair of parents, referred to as “crossover”. Any currentpopulation can be replaced with children to form a subsequentgeneration.

The algorithm can be set to stop when one or more selected stoppingcriteria are met. Stopping criteria can include but is not limited to: anumber of generations produced, a limit on an amount of time, apopulation fitness value being below a set limit, and/or having nosignificant change over a set number of generations or amount of time.

Controllability of the algorithm parameters can depend on the particularsoftware that used to perform the algorithm. In some embodiments hereindescribed, MATLAB® Global Optimization Toolbox software is used. In thissoftware, almost all of the algorithm parameters can be controlled (seeonline documentation for full details [13]).

According to some embodiments, the algorithm parameters can becontrolled based on a desired outcome of the algorithm. These parametersinclude but are not limited to factors as mutation rate, cross-overrate, elite/parent selection, seeding algorithm, number of populationsand subpopulations to use, and termination conditions. Such factors canaffect the speed and accuracy of the genetic algorithm optimization.Adjustments to these parameters can be made in the modifying thedefinition of the genetic algorithm. For example, an algorithm can beseeded in order to input a solution (e.g. a configuration of turbines)which is known and/or expected to be a good solution (e.g. aconfiguration of turbines having one or more of a desired attribute) asa starting point. The parameters used in the genetic algorithm candetermine the method by which the variables are searched over (e.g.having a method which can have a higher computational speed or having amethod which can provide better accuracy). For example, a highermutation rate can mean that there are more jumps from one solution toanother rather than small improvements. Therefore, increasing themutation rate can help to avoid getting caught in a more locallyoptimized solution, rather than a more globally optimized solution.Therefore, there can be trade-offs in changing each of the geneticalgorithm parameters, which can affect how genetic algorithm progresses.

In some embodiments, the method providing a low-order potential flowmodel according to the present be a computer-implemented method, forexample a computer-implemented method, which can execute, using on morecomputer systems, executable instructions to perform one or more of thesteps herein described. For example, the steps can be implemented inhardware, software, firmware or combination thereof.

Features described as blocks, modules or components can be implementedtogether (e.g., in a logic device such as an integrated logic device) orseparately (e.g., as separate connected logic devices).

The methods according to the disclosure can be performed, at least inpart, on a single computer (See e.g. FIG. 23A) or by parallel processingwith two or more computers as shown schematically in FIG. 23B.

A software portion of the methods of the present disclosure can comprisea computer-readable medium which comprises instructions that, whenexecuted, perform, at least in part, the described methods. Inparticular, according to some embodiments, software that is capable ofimplementing a genetic algorithm can be used, for example, MATLAB®,SCILAB®, and Octave. According to some embodiments, the geneticalgorithm can be written by a user, for example using a programminglanguage. Programming languages suitable for performing the geneticalgorithm according to embodiments herein described include but are notlimited to C/C++, Java, Python, and Fortran.

In some embodiments, the instructions, when executed, perform at a stepof receiving acquired flow data around a selected VAWT, for example, byreceiving input from a user.

In some embodiments, the instructions, when executed, perform a step ofadopting one or more of the selected VAWT to serve as a model. The stepof adopting one or more of the selected VAWT can be performed, forexample, based on input from the user.

In some embodiments, the instructions, when executed, perform a step ofassigning one or more variables corresponding to a set of potential flowelements, for example, a rotational (Γ) potential flow element, ablockage effect (μ) potential flow element, a potential flow elementrepresenting energy taken in by the VAWT (q or q_(sink)), and apotential flow element representing turbulence (q or q_(source)). Inthese embodiments, the variables can be input by a user or can beperformed by the software.

In some embodiments, the instructions, when executed, perform a stepapplying an algorithm to the one or more variables. In theseembodiments, the algorithm is configured to output values for each ofthe one or more variables which match the acquired flow data around theselected VAWT according to a fitness function to provide values for eachof the one or more potential flow elements representing potential flowaround the selected VAWT. In these embodiments, the fitness function canbe input by a user or can be selected by the software from a pluralityof set fitness functions included in the software.

The computer-readable medium can comprise, for example, a random accessmemory (RAM) and/or a read-only memory (ROM).

The instructions can be executed by a processor (e.g., a digital signalprocessor (DSP), an application specific integrated circuit (ASIC), or afield programmable logic array (FPGA)).

By way of example and not of limitation, as shown in FIGS. 22A-B, themethods of the present disclosure can be implemented in a computersystem (30) and more particularly, can be implemented in computer system(31) comprising a memory (32), a processor (33), a communicationsinterface (34) and an interconnection arrangement (35) coupling thememory, processor and the communications interface, wherein the memoryis encoded with instructions for executing one or more steps of themethods of the disclosure.

The present disclosure allows, in some embodiments, a method forconfiguring an array of vertical axis wind turbine (VAWTs) for an arraysite based on a desired one or more attributes of the array, forexample, based on power generated by the array per unit land area, anenvironmental impact, social, acoustic, visual impacts, or radarpresence.

In some embodiments the method of configuring the array can be acomputer-implemented method, for example a computer-implemented method,which can execute, using on more computer systems, executableinstructions to perform the steps herein described.

The method comprises providing data corresponding to potential flowaround a selected VAWT. In embodiments wherein the method iscomputer-implemented, the method can execute, using on more computersystems, executable instructions to perform a receiving of input datacorresponding to potential flow around the selected VAWT. The inputdata, for example, can be from the low order potential flow modelaccording to embodiments herein described.

The method further comprises selecting one or more input variables, theone or more input variables representing at least one of a parameter ofthe array site, a constraint of the array site, a parameter of the VAWT,and a constraint of the VAWT. Parameters of the VAWT can include but arenot limited to, diameter, a number of blades, rotational speed, andoperative incoming wind speeds. Constraints of an array sites caninclude, but are not limited to, where the turbines can be positioned,the topography of the array site, and incoming wind flow directions andpredominance. Other constraints of the array site are identifiable by askilled person.

In embodiments wherein the method is computer-implemented, the methodcan execute, using on more computer systems, executable instructions toperform a receiving of the one or more input variables representing theat least one of a parameter of the array site, a constraint of the arraysite, a parameter of the VAWT, and a constraint of the VAWT.

The method further comprises selecting a number of VAWTs to beconfigured in the array. In embodiments wherein the method iscomputer-implemented, the method can execute, using on more computersystems, executable instructions to perform a receiving input concerninga number of VAWTs to be configured in the array. The number of VAWTs tobe configured in the array can be selected simply based on a numberavailable or, for example, based on space available in an array site.

The method further comprises receiving input concerning one or more of adesired attribute of a configured array site. In embodiments wherein themethod is computer-implemented, the method can execute, using on morecomputer systems, executable instructions to perform a receiving ofinput concerning one or more of a desired attribute of a configuredarray site. For example, the array site can be configured for poweroutput.

In some embodiments, the method further comprises designing a fitnessfunction configured to rank configurations of array sites within one ormore constraints of the array site. For example, the fitness functioncan be designed to evaluate each possible array configuration withrespect to set criteria and from this evaluation a quantified rank ofhow good the array performs with respect to these criteria can begenerated. As a further example, the fitness function can rankconfigurations of each possible array site according to a thresholddetermined by a user. A threshold can be set according to a desiredattribute of a configured array site, for example, power output.

In other embodiments, the method of configuring the array can be amethod based on recursive manual computations. Such a method can in someembodiments take substantially longer to perform compared to computerbased computations.

In some embodiments, a fitness function can be written such that the sumof the wind incoming into each turbine (Σ_(i=1) ^(k)u_(∥)) is maximized.The fitness function can be described as a set of rules which can beconfigured to reward one or more attributes of a system which areconsidered to be desirable by a user with a high quantified rank andgiving a lower quantified rank to a system having the one or moredesirable attributes to a lesser extent and/or having one or moreattributes considered to be undesirable by the user. Therefore a fitnessfunction can be controlled in order to optimize, at least locally, for acombination of desired attributes.

The fitness function, according to some embodiments, can also bedescribed as a function that takes a position of a turbine in an arrayas input, operates on that input in a set manner, and outputs a value,for example, a single number. In these embodiments, the number, inconnection with other numbers or values obtained for a particularconfiguration of turbines in an array, can be used to rank theconfiguration of turbines in the array and compare to otherconfigurations. Therefore, changing the fitness function from oneoptimization of a configuration of turbines to another optimizationcomprises altering the operations performed on the input. For example,the fitness function can be set to penalize (e.g. by subtracting a setamount from the value to be output by the fitness function) aconfiguration of turbines which is considered by a user to have aundesired attribute and/or to reward (e.g. by adding a set amount fromthe value to be output by the fitness function) a configuration ofturbines which is considered by a user to have a desired attribute.

In embodiments herein described, a user can select any one of the rankedconfigurations, for example, to be used in arranging an array ofturbines. The user can selected a highest ranked configuration, whichcan be considered to be a locally optimized configuration, or the usercan select another configuration among the ranked configurations, whichcan be considered as not being locally optimized. Methods for arrangingturbines in an array will be understood by a skilled person.

By way of example and not of limitation, if a desired array site is onthe side of, and including the crest of a hill, it can be desired tooptimize both for power output of the turbine array and for keepingturbines off of the crest line in order to reduce visual impact. In thisexample, the fitness function, for each possible array, can be set tosum the wind incoming to each turbine (cubically proportional to power)and can also be set to subtract a fixed amount for each turbine on thecrest line. Depending on what the fixed amount is set to be subtractedfor each turbine on the crest line is, a balance can be found between ahigh power output and a minimal visual impact.

As another non-limiting example, if it were known and/or found that asolid line of closely-spaced turbines is difficult for birds and bats toavoid while flying, the fitness function can penalize by a set amount, aconfiguration of turbines that comprises the solid line of turbines incomparison to a configuration of turbines that does not comprise thesolid line of turbines. In this example, the fitness function can be setto subtract a fixed amount for any such the solid line of turbines in anarray configuration. In some embodiments, a configuration of turbineswhich is penalized in this way, can still be balanced against anotherdesired feature, such as a low radar profile.

As further non-limiting example, if it is known and/or found that acertain turbine configuration provides an acoustical resonance thatdisproportionally increases the acoustic signature that turbineconfiguration, a fitness function can penalize an array which comprisesthat configuration within the overall configuration of the array. Forexample, if a three-turbine triangular configuration exhibits acousticalresonance that disproportionally increases the acoustic signature of thethree turbines, the fitness function can be designed to penalize anarray having the turbine triangle. In particular, the fitness functioncan be designed to penalize the turbine triangle in accordance with theposition of the turbine triangle with respect to a residential area. Inparticular, the fitness function can be designed to increase the penaltywith a decreasing distance of the turbine triangle from the residentialarea and accordingly decrease the penalty with an increasing distance ofthe turbine triangle from the residential area. In this way, the arraywould be quiet in regions where acoustics mattered, but can be morepower efficient in regions where acoustics did not matter.

For design of a fitness function for configuring an array of verticalaxis wind turbines (VAWTs) according to some embodiments, it can beestimated that P∝u_(∥) ³ and the power coefficient, C_(p) can be defined[5] as:

$\begin{matrix}{C_{p} = \frac{P}{\left( {1/2} \right)\rho \; {AU}_{\infty}^{3}}} & (14)\end{matrix}$

where P is the power output of the turbine, ρ is the air density, A isthe area swept by the rotor (equal to turbine diameter times turbineheight), and U_(∞) is the free-stream wind speed. Extending thisanalysis to an array of k turbines yields:

$\begin{matrix}{C_{p,{array}} = \frac{\sum\limits_{i = 1}^{k}\; \left( P_{i} \right)}{\left( {1/2} \right)\rho {\sum\limits_{i = 1}^{k}\; {\left( A_{i} \right)U_{\infty}^{3}}}}} & (15)\end{matrix}$

Similarly, in order to compare the performance of an array of k turbinesto the performance of k isolated turbines, the normalized coefficient(denoted for simplicity as C_(p)′) according to some embodiments can beconsidered to be:

$\begin{matrix}{C_{p,{array}}^{\prime} = {\frac{C_{p,{array}}}{k*C_{p,{iso}}} = \frac{\sum\limits_{i = 1}^{k}\; P_{i}}{P}}} & (16)\end{matrix}$

In embodiments wherein the method is computer-implemented, the methodcan execute, using one or more computer systems, executable instructionsto perform a receiving of input of a fitness function configured to rankconfigurations of array sites within one or more constraints of thearray site, according to one or more set thresholds, the thresholdsbeing set according to a desired attribute of a configured array site.

In some embodiments, the method further comprises applying an algorithmto the one or more variables, the algorithm configured to output datacorresponding to locations of VAWTs, which provide the one or more of adesired attribute of the array of VAWTs. In embodiments wherein themethod is computer-implemented, the method can execute, using one ormore computer systems, executable instructions to apply an algorithm tothe one or more variables, the algorithm configured to output datacorresponding to locations of VAWTs which provide the one or more of adesired attribute of the array of VAWT units.

In some embodiments, in order to provide configurations of turbinearrays based on a desired one or more attributes of the array, a geneticalgorithm which allows for optimization of variables over integers (e.grather than over the real numbers) is used, for example, MATLAB® toolboxGEATBX® [9]. Other algorithms suitable for methods herein described areidentifiable by a skilled person upon reading the present disclosure.

Configurations of turbine arrays herein described can be provided inconsideration of a single prevalent airflow or in consideration of morethan one prevalent airflow.

In some embodiments, the methods herein described can be used tooptimize configurations of turbines at an array site for which there ismore than one primary wind flow direction. For these embodiments, anarray configuration can be ranked by the optimizing algorithm's fitnessfunction based on its performance at each of the one or more primarywind directions. The array that performs the best over all of the one ormore primary wind flow directions can then be ranked the highest. Theseembodiments can further take into account an importance of each winddirection. For example, if there are primary and secondary and/or lowerimportance incoming wind directions, the relative importance of each canbe incorporated as a weighting of the fitness function. For example, theperformance of the array for the one or more primary wind directions canbe set to influence the ranking of the array to a greater degree thanthe performance of the array for the secondary and/or lower importancewind direction.

In some embodiments, the methods herein described can be used to predicthow larger arrays of turbines will perform, for example, in terms ofpower output, compared to a smaller array, the power output of thesmaller array being determined either experimentally or by simulation.For example, power output of a larger array can be determined based onthe power output of an array having a smaller number of total turbines,but having a same arrangement of co-rotating or counter-rotating pairs,a same turbine-turbine spacing, and a same pair-pair spacing.

In some embodiments, a homogeneous strength model is used in predictingperformance of turbine arrays and in others an iteratively definedstrength model is used (See e.g. Examples 5-8).

In the homogenous strength model each of the turbines in an array can beassigned a same value of each of Γ, μ, and q. In the iteratively definedstrength model, each turbine in the array can be assigned a value ofeach of Γ, μ, and q according to a respective location of each turbinein the array.

This iteratively defined strength model can be used in embodiments wherea higher accuracy is desired. For example, particularly in the case ofproviding configurations of turbine arrays comprising more turbines thana number of turbines from which the model is derived, the iterativelydefined strength model can be more accurate (See e.g. Example 7). Thehomogeneous strength model can be used in embodiments where shortercomputation times are desired.

In embodiments where it is desired to provide configurations of turbinearrays comprising a number of turbines which is close to the number ofturbines on which the model is based, there homogeneous model canprovide accuracy substantially equivalent to that of the iterativelydefined strength model. In some embodiments, the number of turbines usedin the prediction can be double or triple the number used to develop thehomogeneous strength model.

In embodiments where Γ, μ, and q are modeled using simulated data, thehomogeneous strength model can be used with substantially the sameaccuracy as the iteratively defined strength model (See e.g. Example 8).In embodiments where Γ, μ, and q are modeled using experimental data, aselection between the iteratively defined strength model and homogeneousstrength model can be based on a desired computation time, a desiredaccuracy level, and/or a compromise between the two.

In some embodiments, in order to provide shorter computation times, theiteratively defined strength model can be used in conjunction with anextrapolation to determine performance of large arrays (See e.g.Examples 7-8). For example, the iteratively defined strength model canbe used to predict a set of different array sizes and the data from eachcan then be normalized and extrapolated into much larger array sizes.

By way of example, and not of limitation, a centerline velocity can becalculated for 3×3 array, a 4×4 array, a 5×5 array, a 6×6 array, a 7×7array, an 8×8 array, and a 9×9 array using the iteratively definedstrength model. The centerline velocities can then be plotted and fittedto a trend line for extrapolation.

In some embodiments, a linear equation can be used to fit the centerlinevelocities. In these embodiments, a least squares analysis can be usedto estimate the accuracy of such extrapolation (See e.g. Example 7,FIGS. 9A-B).

In embodiments where Γ, μ, and q are modeled using simulated data,configurations of large array sizes can be provided relatively quickly.For example, optimized array configuration for arrays of 100×100turbines can be calculated in less than a minute.

In embodiments where the method of configuring an array is acomputer-implemented method, the method can be performed, for example,on the same computers and systems described herein with reference to themethod of providing a low-order potential flow model.

In these embodiments, the instructions, when executed, perform a step ofreceiving input data corresponding to potential flow around a selectedVAWT. The input data can be, for example, the values for each of the oneor more potential flow elements representing potential flow around theselected VAWT obtained from the method of providing a low-orderpotential flow model. The data can be input by a user or the software,if used in conjunction with the computer-implemented method of providinga low-order potential flow model, the software can be configured toretrieve and receive the data provided in the method of providing alow-order potential flow model.

In the computer-implemented method for configuring the array, theinstructions, when executed, perform a step of receiving one or moreinput variables, for example, input variables representing a parameterof the array site, a constraint of the array site, a parameter of theVAWT, and/or a constraint of the VAWT.

In some embodiments, the instructions, when executed, further perform astep of receiving input concerning one or more of a desired attribute ofa configured array site.

In some embodiments, the instructions, when executed, further perform astep of receiving input of a fitness function configured to rankconfigurations of array sites within one or more constraints of thearray site, according to one or more set thresholds, the thresholdsbeing set according to a desired attribute of a configured array site(18.2);

In some embodiments, the instructions, when executed, further perform astep of applying an algorithm to the one or more variables (18.3)configured to output data corresponding to locations of VAWTs whichprovide the one or more of a desired attribute of the array of VAWTs

EXAMPLES

The methods and systems herein disclosed are further illustrated in thefollowing examples, which are provided by way of illustration and arenot intended to be limiting.

Example 1 Obtaining Experimental Data

Eighteen turbines, nine counter-clockwise rotating and nine clockwiserotating were arranged in an array of counter-rotating pairs, spacedabout 8 turbine diameters apart in each direction, as shownschematically in FIG. 1. In this example, the turbines were modifiedversions of a commercially available model from WINDSPIRE® Energy Inc.Each turbine was 10 m tall, had a diameter of 1.2 m and was connected toa 1200-W generator. The cut in wind speed of the turbines was 2.8 m s⁻¹.The turbine array was located in Antelope Valley of northern Los AngelesCounty, California, USA. The site is desert and the topography is flatfor approximately 1.5 kilometers in all directions [5]. FIG. 1 shows aschematic of the field site.

The electrical power generated by each turbine was monitored inreal-time and recorded at 1 Hz using power meters connected to a centraldata logger (Campbell Scientific Inc.). Measurement accuracy was +/−5%.Additionally, a 10-m meteorological tower was consecutively placed ateach of the gray squares in FIG. 2A. This diagonal through the array isreferred to as the centerline of the array. The tower was left in eachposition for approximately 10 days. The accuracy of the 3-componentsonic anemometer (Campbell Scientific Inc., CSAT 3) was +/−1%. Data fromthe meteorological tower were also recorded at 1 Hz using a datalogger(Campbell Scientific). The data was taken over four months over whichtime the average wind speed was approximately 7 m/s. Since thepredominant wind direction at this site is from the south-west, thelower left hand corner of the array as presented in FIG. 1 was taken tobe the origin of the coordinate systems as indicated.

In relating the power output of the turbine (P) to the flow conditions,it was estimated that P∝u_(∥) ³ where u_(∥) is the flow directed intothe turbine (i.e. parallel to the free-stream direction). The measureddata is presented in FIG. 2.

Example 2 Obtaining Simulated Data

In order to obtain a more complete flow field around the VAWTs to whichto match a potential flow model, two-dimensional simulations of a singleVAWT cross-section, a pair of co-rotating turbines, and a pair ofcounter-rotating turbines were completed. In this example, CFDsimulation were obtained. In particular, the CFD software FLUIDICA® [12]was used. This software uses an immersed-boundary Navier-Stokes solverto compute the 2 D flow around arbitrarily defined geometries. In thisexample, the geometry was made to closely resemble the cross-sections ofthe turbines used for the experimental data, which in this example areWINDSPIRE® turbines. Each turbine comprises three blades equally spacedto define a diameter of unit dimension, D. The chord length of eachblade was approximately ⅛ D. The blades were approximated as flat platesat zero angle of attack with respect to the tangent of the defined(virtual) circle.

Before beginning the simulations, an adequate grid size and time stepwas determined to allow convergence of the flow. Five simulations wererun using a single, stationary VAWT cross-section. TheCourant-Friedrichs-Lewy (CFL) condition was used to determine thenumeric stability of the simulation and was defined according toequation 13 as described herein.

For the simulation to be numerically stable, it is typically required inthis example that CFL≦0.7. For the WINDSPIRE® geometry [12], allsimulations in this example were run at CFL=0.25 and the absolute valuesof Δt and h were varied so that this ratio would hold. Thenon-dimensional values of h chosen were 1.27*10⁻², 6.35*10⁻³, 3.18*10⁻³,1.58*10⁻³, and 7.94*10⁻⁴. The maximum dimensional velocity of the systemwas conservatively taken to be the free-stream velocity plus the tipvelocity of the rotating turbines plus a margin of error of around 0.5m/s. In order to analyze convergence, the forces on one of the blades ofthe turbine were examined over time for each of the five cases. Theresult for the decomposition of the force parallel to the direction ofthe free-stream is given in FIG. 3.

Based on these results and in order to limit computational time, thesimulations performed in this example were conducted using h=3.18*10⁻³.

The final WINDSPIRE® simulations were run with two grid levels—a finegrid close to the turbine and a coarse grid over the entire simulateddomain. The simulated area of the fine grid was taken to be 0.5 D infront of and to either side of the turbines and 5 D downstream of theturbines. The coarse grid was substantially larger: just over 3.5 Dupstream, 1.5 D to either side of the turbine and about 8 D downstream.The simulations were run until the non-dimensional time of t=10. Anexample of the flow field as simulated is given in FIG. 4. In FIG. 4,the free-stream flow is moving in the positive x direction.

Example 3 Potential Flow Modeling

The potential flow models were generated from a set of elementsincluding vortices, dipoles, and sources/sinks. Different numbers andarrangements of these elements were optimized for each model asdescribed in Examples 4-6. The complex potentials (F) and the complexvelocities (W=dF/dz) of the potential flow elements were calculatedusing equations 8-11 as already described herein and the velocity flowfield u was calculated with equation 12.

Example 4 Potential Flow Modeling of a VAWT Cross-Section: GeneralApproach Used in Examples 5 and 6

In order to generate a model, the number of each type of element, theposition of each element, and the strength of each element in thepotential flow was optimized such that the resulting flow fieldu_(model) matched as closely as possible the original (acquired) data,either u_(exp) (the acquired experimental data) or u_(sim) (the acquiredCFD simulated data). Even if some of these variables are fixed prior tothe model parameter optimization, the number of variables can still besignificant, resulting in a multidimensional optimization problem. Agrid search algorithm rapidly becomes computationally unfeasible inthese examples as the number of variables increases. For this reason,genetic algorithm searches were used in these examples to perform themodel parameter optimizations.

For these optimizations, the MATLAB® Global Optimization Toolboxsoftware was used. The following outline, drawn from MATLAB®documentation [13] summarizes how the genetic algorithm works:

The algorithm begins by creating a random initial population. Anindividual is the set of numbers, one for each variable in the problem).A population is a group of individuals).

The algorithm then creates a sequence of new populations. At each step,the algorithm uses the individuals in the current generation to createthe next population. To create the new population, the algorithmperforms the following steps: scoring each member of the currentpopulation by computing its fitness value (using a fitness functionwritten by the user specifically for the optimization); scaling rawfitness scores to convert them into a more usable range of values;selecting members, called parents, based on their fitness; choosing“elite” individuals in the current population that have lower fitness;passing elite individuals on to the next population; producing childrenfrom the parents. Children are produced either by making random changesto a single parent—mutation—or by combining the vector entries of a pairof parents—crossover; replacing the current population with the childrento form the next generation.

The algorithm stops when one of the stopping criteria is met: maximumnumber of generations or the time limit has been reached, populationfitness value is below some predetermined limit, or there is nosignificant change over a predetermined number of generations or timelimit.

Within the scope of software used in this example, almost all of thealgorithm parameters are controllable (See online documentation for fulldetails [13]). After examining the performance of some customizations,it was determined that the default parameters performed comparably andtherefore optimizations in Examples 5-6 were run at default settings.The fitness function, however, was individually written for eachoptimization and is discussed in more detail in Examples 5-6. Thisapproach can lead to non-determinism of the algorithm. Therefore, inthese examples, in order to at least partially compensate for thenon-determinism, multiple iterations of each optimization program wererun as a partial check on the answers.

Example 5 Matching Experimental Data

Ten different potential flow models were optimized and compared. Thequality of each model was taken to be the root-mean-square differencebetween the experimental data points and the corresponding model datapoints. Thus in this example, the fitness function of the geneticalgorithm was set to minimize this RMS difference. In matching thecenterline data velocities from the meteorological tower measurements,the models were analyzed at points corresponding to the experimentaltest points and directly compared. In matching the experimental turbinevelocity data (which was itself taken to be the cube root of themeasured power data), a line of data points was taken perpendicular tothe global free-stream direction and spanning from one edge of theturbine to the other. The data points were of adjustable distance fromthe turbine center, but in this example, the test points were placed 0.6turbine diameters in front of the turbine center. The component of thepotential flow perpendicular to this line (parallel to the globalfree-stream direction) was then determined at each of the test pointsand averaged in order to determine the flow velocity into each turbineas a single number, u_(∥). This analysis is shown in FIG. 5.

For all models examined, a reduction of the number of variables waspossible. All models in this example estimated that there was a vortexand a dipole at the center of each turbine. The models used here wereeither for a single turbine unit where the parameters of a singleturbine were determined such that repeating that model at each turbineposition yielded the best overall results or for a two-turbine unitwhere the parameters of a counter-rotating turbine pair was determinedsuch that repeating that model at each turbine pair yielded the bestoverall results. In both cases, the number of vortices and dipoles wereequivalent to the number of turbines included in each unit.

The sinks were either “free-floating” such that the position wasoptimized within a small box centered on the turbine(s) or were “fixed”such that the position was either predetermined or determined by modeledflow. The sources were either modeled as free-floating (within the samerange as the sinks) or modeled as a source field over the entire arrayconsisting of sources of equal strength placed in a regular grid (forthese optimizations, the sources were separated by 0.25 D in eachdirection and extended several D beyond the edges of the array in alldirections.)

Physically, the vortex element models the rotational componentintroduced to the flow by the rotating turbine. The dipole accounts forblockage effects of the turbine itself. The sinks account for theextraction of energy from the wind by the turbine, while the sourcessimulate flow recovery due to entrainment of free stream flow in thewake. Perhaps more physically realistic than a single source is theapplication of a source field to simulate the inflow of air from abovethe turbine array [4]. For these models, a uniform field was assumed. Inthis example, the sink was placed 2 D downstream of each turbine centerbased on examination of the velocity profile for a single isolated VAWT[1].

The best performing model in this example using a single turbine unitwas composed of a vortex and dipole at the center of each turbine, asink 2 D downstream of the turbine center and a source field. This modelyielded an RMS difference between all the data (both centerline andinflow) and the model of 0.66%. If the strengths of the vortex, dipoleand sink are fixed to a single value for the entire array, the model wasnot able to directly extend in a reasonable fashion to larger arrays.The predicted flows for the larger arrays showed apparently lineardecreases in the centerline flow velocity to values below zero. Thisresult is shown in FIG. 8. A modification was therefore made to thealgorithm. The strengths of the vortices and sinks for each turbine wereestimated to be proportional to the cube of the incoming velocity; thatis, if the incoming flow velocity to turbine j is given by v_(j), thenΓ_(j)∝av_(j) ³ and q_(j)∝bv_(j) ³ where a,b are some constants ofproportionality. Similarly the strength of the dipole for each turbinewas estimated to be proportional to the square of the incoming velocity:μ_(j)∝cv_(j) ², where c is some constant of proportionality. The cubicproportionality of the vortex was estimated because ideally the vortexcaptures the rotation of the turbine which is proportional to the poweroutput of the turbine which is proportional to the cube of velocity.Similarly, the sink ideally represents the wind power converted toelectricity, which is also proportional to the power output of theturbine and therefore is also proportional to the cube of velocity. Thedipole ideally represents the blockage effect of the turbine or thedrag, which can be taken to be proportional to the square of thevelocity. Using these proportionalities, the four variables of the modelare then three constants of proportionality and the strength of thesources in the field. This process requires iteration betweendetermining the strengths of the elements and the flow field velocities.Thus, this model is more computationally intensive than the homogenousstrength model, however, in this example, it results in more realisticpredictions for larger arrays—the centerline velocities dropped at anon-linear rate rather than a linear rate as distance from the beginningof the array increased and zero values were not attained. The RMS overall data points of model 7 was 0.69%.

Example 6 Matching CFD Data

The first step in the analysis of the CFD data was to average the dataover the simulated time span in order to obtain a time-averaged flowpattern. Although a time-varying potential flow model can be used, itwas believed that the response time of downstream turbines is longerthan that of the short-term flow variations, therefore allowing thetime-averaged flow field to be a good approximation of the flow in thearray for this example. The next step in the analysis was to down samplethe CFD data to lower spatial resolution, because it would have beencomputationally unfeasible to consider every grid point used in the CFDanalysis in the final optimization. The resulting flow field for asingle WINDSPIRE® turbine is given in FIG. 6 in which the uniformapplied flow is moving in the positive x-direction.

FIG. 6 shows a higher density of data points to be matched in the nearvicinity of the turbine. These points have been drawn from the fine CFDgrid while all other points have been drawn from the coarse grid. Thecolor variation of the flow vectors indicates how the flow field wasdivided for modeling purposes. After multiple attempts to model the flowfield with a single potential flow model, the results were notsatisfactory. Therefore the wake region and surrounding region of theflow field were modeled with separate potential flow models, albeit forconsistency the same type of model (i.e. same number of each type ofelement) was used for both regions with only the fitness functionschanged appropriately. As with the experimental data, the number ofvariables was reduced by assuming one vortex and one dipole at thecenter of each turbine. Again, “free-floating” indicates that thepositions of the element can be optimized within a small box surroundingthe turbine. In these cases the box extended 1.5 D in the upstream(negative x direction) and in the directions perpendicular to the flowdirection (positive and negative y-directions) and 2 D in the downstreamdirection, where all distances are measured from the turbine center.

The fitness functions for the surrounding region and the wake regionwere slightly different in order to emphasize the differentcharacteristics of the flow deemed important. Most importantly, anydifferences between the simulation and modeled flow velocities in thefar downstream (e.g. where another turbine might be placed) wereweighted to contribute more to the overall cost of the tested model thanflow velocity differences in other regions of the flow. The weightingsof the fitness functions were also designed to favor models thatunderestimated the downstream flow rather than overestimate it, in orderto provide a conservative model.

The goodness of fit of a model was determined by taking the RMSdifferences (for each flow direction) between the average at crosssectional slices of the simulated field and the modeled field. Thecross-sectional slices can be across the entire field, the band in frontof and behind the turbine, and just the centerline of the system. Inthis example, the type of model used for both the experimental modelingwhich comprised a source field and fixed sink was not able to provide asgood of a match as the source and sink free-floating cases. Thedifference in optimal modeling can be due to the discrete nature of thesource field. Although the ideal field would be a continuous, uniformfield, with the potential flow elements in use, only source terms atdiscrete locations can be used—in this case 0.25 D spacing in bothdirections. Although this method would work for the sparse data settaken experimentally, with the number of data points used in the CFDset, the positions of the source elements can lie very close to sometest points and thus skew the flow. The net result would be that onedisruption in the overall flow (i.e. a single free floating source)would provide a better fit to the CFD data than many sources each addinga disruption to the overall flow pattern.

Models using a vortex and dipole at the center of each turbine and threefree-floating sources/sinks (11 variables for a single turbine, 13variables for turbine pairs) provided the overall best fits between thesimulated and the modeled flow field in these examples and was thus usedalso in Example 8. A comparison of the average flows at cross-sectionalbands for the simulated and modeled flows is provided in FIG. 7 for thesingle turbine.

Example 7 Results of an Experimental Data Model

In this example, it was estimated that P∝u_(∥) ³. The power coefficient,C_(p) was estimated with equation 14 as described herein. Rather thanconsidering the absolute value of C_(p), in these examples, thecoefficient was normalized by dividing by the corresponding coefficientfor a single isolated turbine (denoted C_(p,iso)) in order to yieldC_(p)′. Extending this analysis to an array of k turbines providedequation 8, as also described herein. Similarly, in order to compare theperformance of an array of k turbines to the performance of k isolatedturbines, the normalized coefficient (denoted for simplicity as C_(p)′)was estimated with equation 9, also described herein.

C_(p,array)′ is used in this example as the primary means to comparedifferent array configurations.

The first application of the experimentally-matched potential flow modelwas to examine the flow patterns of a larger version of the same type ofarray as that tested, i.e. the same counter-rotating pairs arrangementswith the same turbine-turbine spacing and the same pair-pair spacing,but with a larger number of total turbines. For simplicity, these arraysare referred to by the number of turbine pairs there are on each side ofthe array (i.e. the experimental array is a 3×3 array.) The velocityprofile along the centerline through the array was taken as acharacterizer of the arrays.

As previously mentioned, the homogenous strength model performed well inmatching the experimental data, however when the model was applied tomuch larger arrays, the results were deemed unphysical. Namely, thecenterline velocity profile continued to decrease in magnitude from thefront to the back of the array, and in very large arrays would even gonegative (as shown in FIG. 8, left.) It is therefore not expected thatthe turbines where the flow is negative would continue to rotate withthe same speed or even in the same direction as those at the front ofthe array and therefore the model used at the front of the array isexpected not to be the same as that towards the back of the array.Therefore, the iteratively defined strengths model was developed, inwhich the strengths of the vortex, dipole, and sink of each turbine isdirectly dependent on the wind speed incoming into the turbine. Asmentioned previously, however, this model required iteration between thestrengths of the vortices, dipoles, and sinks and the velocity into eachturbine in order for a converged, equilibrium state to be reached.Furthermore, if the initial guess as to the incoming flow velocity istoo far off of the converged state, the model can become unstable anddiverge from an equilibrium state. These requirements can be limiting inthe size of an array that can be computed via this model. However, forwhat array sizes were computed in these examples, the centerline profileappears to be much more intuitively correct as the centerline velocitydecreases at a decreasing rate from the front of the array to the back.In FIG. 8 (b), the homogenous strength and iteratively defined strengthmodels are compared for a 9×9 array.

For arrays larger than 9×9, the iteratively defined strength modelproved to be computationally intensive. It was therefore whether anyrelationships in the performances can be determined from the smallerturbine arrays and can allow an extrapolated prediction of the largerarray performance. For this purpose, the iteratively defined strengthmodel was used to compute the centerline velocity profiles for allsquare array sizes 3×3 to 9×9. The results were normalized by thecorresponding dimension of the array along the centerline. Plotting allresults yielded FIG. 9A. The dashed lines in FIG. 9A indicates thepoints at which specific values were taken from each of the arrays, inparticular, one array length in front of the arrays, one turbinediameter in front of the arrays, one turbine diameter behind the arrays,and one array length behind the arrays. Because the centerline profileswere normalized to an array diameter, the relative size of one turbinediameter upstream/downstream changed for each array size, thus thedashed lines indicating these points are angled in FIG. 9A. Evaluatingeach array size at these marked points yielded the results shown in FIG.9B and the data are fit with a linear equation. The R² values of thefits are 0.92, 0.97, 0.94, and 0.86, respectively. A nonlinear fit ofthe form y=a−be^(−cx) was also attempted. The fit for the 1 turbinediameter upstream data and the fit for the 1 array length downstreamwere somewhat better than for the linear curve: R² values of 0.98 and0.96 respectively, while the other two curves, the one array lengthupstream and the one turbine diameter downstream were essentially notfit, R² values of <0.001. In order to determine the more appropriatecurve fit, data for larger arrays can be used.

From these results, it was observed that the increased incomingvelocities with increasing array size caused the turbines at the frontof the array to have higher individual power outputs. However, since thedownstream velocities decrease with increasing array sizes, it can beconcluded that the turbines in the far downstream are operating at alower power output. Since the rate of increase in the upstream flows isapproximately the same as the rate of decrease in the downstream flows,it can be estimated that the overall C_(p,array)′ will stay relativelyconstant as the number of turbines in the array is increased.

Another application of the developed model was to allow optimization ofarrays of turbines. In this example it was desired to optimize thecurrent field setup so that any final predictions can be experimentallyverified. Thus, the optimization in this example was bounded byrequiring 9 clockwise turbines and 9 counter-clockwise turbines be usedand the turbines can only be located at discrete positions as indicatedin FIG. 1. For these optimizations, the independently written geneticalgorithms MATLAB® toolbox GEATBX® [9] was employed because it allowedoptimization of variables over the integers rather than over the realnumbers (unlike the MATLAB® Global Optimization Toolbox). The algorithmsused by this software are the same as for the MATLAB® GlobalOptimization Toolbox, discussed previously. The fitness function of theoptimization was written such that the sum of the wind incoming intoeach turbine (Σ_(i=1) ^(k)u_(∥)) was maximized.

The array was optimized in this example for a single prevalent incomingflow direction, in this case, the southwest. The optimal arrangement wasdetermined to be a “V” shape open to the incoming wind direction, asshown in FIG. 10. In this arrangement, a minimal number of turbines aredirectly in the wakes of other turbines. Note that all thecounterclockwise turbines are grouped together on one side of the V andall the clockwise turbines are located on the other side of the V suchthat all turbines rotate from the outside to the inside of the V.Because co-rotating turbines are placed closely together in a line forthis arrangement, it was expected that the airflow would be channeledalong the inside and outside of the lines of turbines in such a way asto move with the turbines rather than pushing against them. Essentially,vortex sheets of turbines are created on either side of the V.

The performance of such an array when the airflow is not in thedirection for which the array has been optimized was also considered.Computing the C_(p,array)′ value for the given array at intervals of0.007 radians for the entire 360° range of possible incoming flowdirections yielded the C_(p,array)′ verses flow direction curve found inFIG. 11. Note that in reference to the field array presented in FIG. 1,0° corresponds to wind incoming from due north.

FIG. 11 shows that over a range of incoming flow directions ofapproximately 30° centered on the incoming flow angle for which thearray was optimized, the array performed about 3.5 times better than thecorresponding number of spatially isolated turbines. Over a range ofapproximately 90° degrees, centered on the optimized angle, the arrayperformed at least 3 times better than an isolated array. It isimportant to note, however, that in a region spanning approximately 90°,centered on the incoming direction that is 180° from the optimizeddirection, the array performs worse than an isolated array. In FIG. 11,the solid line, indicates the computed C_(p,array)′ for each angle isvery noisy. As described herein, for the iteratively defined strengthmodel, an initial guess for the incoming wind can be made and then thestrengths of the potential flow elements and the velocity field can beiterated over. The method by which the program makes an initial velocityguess and performs the ensuing iteration is non-deterministic. Thus somesharp variations from one angle to another can be present. The filteredcurve presented in FIG. 11 likely can be considered to be a morereasonable approximation of the actual performance of the array. It isalso noted that in a fully deterministic program, the C_(p,array)′ curvecan be symmetric about the optimized angle due simply to the geometry ofthe array.

An optimization of the array given a uniform distribution of winddirections can also be performed.

Example 8 Results of a CFD Data Model

The first application of the CFD model was to examine the predictedcenterline profile of arrays of counter-rotating turbine pairs tocompare the results to the experimentally based model. The models inthese examples were developed for isolated turbines or turbine pairswith a normalized incoming wind speed. In placing them in an array, twoapproaches were taken: the first approach assumed homogeneous turbinesthroughout the array, while the second approach calculated the flow infront of a turbine due only to the wake of the turbine immediately infront of it. These two approaches and the results of the experimentalmodel for the same sized array are compared in FIG. 12.

There is minimal computational time difference between the twoapproaches to the array evaluation, therefore the second approach, whichassumed influence from directly upstream turbine wakes, was selected forfurther analysis in this example since it proved to be a slightly closermatch to the experimentally derived model predictions. The computationtime for the CFD model was very rapid—typically even the computationsfor arrays as large as 100×100 turbine pairs took only a few tens ofseconds. This is in contrast to the computation time required by theexperimentally derived model which required, in some examples, tens ofminutes to compute arrays as large as 9×9 turbine pairs and it was notcomputationally feasible to compute larger arrays. The time differencecan be due to the experimental model's requirement in some cases foriteration.

With the ability to compute the larger arrays more rapidly, it wasconsidered if any velocity trends became apparent between arrays ofdifferent sizes. Again, arrays of different sizes were computed andnormalized by the array diameter. Comparisons of the velocities werethen considered at set points along the centerline: 1 array diameterupstream, 1 turbine diameter upstream, 1 turbine diameter downstream,and 1 array diameter downstream, as indicated in FIG. 13.

Although exponential fits were attempted for this data, linear fitsproved to have far higher R² values than the exponential fits. A summaryof the R² values for the linear fits and the coefficients of the fitsfor both the experimental and CFD derived models is provided in thetable 1.

TABLE 1 Summary of the R² values for the linear fits and thecoefficients of the fits for both the experimental and CFD derivedmodels. Experimental Model CFD model R² a b R² a b 1 array 1 0.00180.9997 0.92632 0.0036 1.0249 diameter upstream 1 turbine 0.99743 0.00620.9044 0.96961 0.0168 1.0968 diameter upstream 1 turbine 0.99984 −0.00620.993 0.92609 −0.0098 0.9353 diameter down- stream 1 array 1 −0.00180.9984 0.84571 −0.0022 0.9772 diameter down- stream

It is difficult to tell from these data if the overall C_(p,array)′ willincrease or decrease. However, using the CFD model the computation ofC_(p,array)′ for large arrays becomes possible. In this example, arraysof size 3×3, 5×5, 7×7, 9×9, and 15×15 were examined. It can be seen fromFIG. 14 that the value of C_(p,array)′ decreases exponentially with thesize of the array. The trendline fit was of the form y=a−be^(−cx). Thecoefficients were found to be a=0.7635, b=−0.5076, and c=0.1527. The R²value of the fitting was 0.9997. At present it is unknown if it ispossible to extract this information from the centerline trend dataonly.

The CFD model was then also used in a program seeking to optimize aturbine array subject to the same constraints as the correspondingoptimization using the experimental model. For this optimization, thestrengths of the potential flow elements were taken to be homogenousthroughout the array, but the contribution of every element in the arraywas computed for each test point. Using the CFD model yielded the sameoptimized array as the experimental model. Note that the model unitsused included 4 clockwise rotating turbine pairs, 4 counter-clockwiserotating turbine pairs, and 1 clockwise/counter-clockwisecounter-rotating turbine pair. The computed C_(p,array)′ of using theCFD model was 1.5903 at the optimized incoming wind direction. Theimprovement predicted by the CFD model is therefore about half thatpredicted by the experimental model. There were also several otherarrangements of turbines that were found to have performanceimprovements of very similar magnitude. At present, the discrepancybetween the predicted improvements from the two models is unknownalthough it is hypothesized that the difference is due to the inclusionor exclusion of downstream effects.

How the array would perform when the wind was not incoming from theoptimized direction was considered here again in this example. It wasdiscovered that some of the potential flow elements in the turbine pairmodels were very close to lying exactly between the two turbines. Thus,when the potential flow elements were rotated around this center inorder to account for the wind incoming from different directions, theseelements stayed in approximately the same place and the test pointstaken to determine the flow into the turbines came very close to thesepoints. The net result was that the potential flow computed at thesepoints was unrealistically high or low. Therefore, in order to evaluatethe performance of the array when the test points were between twoturbines in a pair (i.e. when one turbine of a pair was in the wake ofthe other turbine in the pair), the models were switched from using theturbine pair models to using the single turbine models. The results ofthis analysis are shown in FIG. 15. It is noted that the nonlineareffects captured in the pair models are important in this example sincethe C_(p,array)′ values predicted by the single turbine models aresubstantially smaller than those predicted by the pair models in theseexamples. This also indicates that for the wind directions at which thepair models could not be used, the C_(p,array)′ values are more likelyto be an underestimate rather than an overestimate. Although not assubstantial as the increases seen in performance predicted by theexperimental model, nevertheless there is improvement over a range ofapproximately 60° around the optimization angle and another peak againof approximately 60° width at the opposite direction of optimized angle.

As a check on these results, the directional flow analysis around asingle co-rotating pair and a single counter-rotating pair using thesingle turbine models was completed. As expected, there was symmetryaround the centerline in the counter-rotating case with the flow intothe turbines from the optimized direction being better than the flowfrom opposite direction. Similarly, there was an inverted symmetryaround the centerline of co-rotating turbine pair. The plots of theseresults are given in FIG. 16.

Examples 1-8 Summary

The examples above show an implementation of embodiments of the loworder potential flow model of the cross-section of a vertical axis windturbine described herein which can be used to predict and optimize thepower outputs of large arrays of such turbines. In these examples, thepotential flow models have been optimized both for a set of experimentaldata and for a set of CFD data.

The CFD modeling procedure, in particular, can be extended to modelturbines of substantially any geometry. The experimental based model canbe extended to model turbines in any array for which the power output ofthe turbines is known. In some cases, knowledge of the velocities withinthe current array is also known. Furthermore, since the elements of thepotential flow model can have physical interpretations, performances ofarrays with arbitrary turbines can be correlated and predicted, giveninformation on how that turbine physically compares with known turbines(e.g. more blades can correspond to a higher dipole value since therecan be more flow blockage).

The two models were used in these examples to empirically determine thescaling laws for the velocities associated with large arrays of equallyspaced counter-rotating pairs of turbines. It is predicted that thevelocities in front the arrays will increase linearly, while thevelocities behind the arrays will decrease linearly. However, comparisonof the C_(p,array)′ values computed using the CFD derived modelindicates that there will be an exponential decrease in the value ofC_(p,array)′ as the number of turbines in the array increases.

Both models were also used in order to predict an optimal arrangementgiven the physical parameters of the field array currently in use andassuming a single prevalent wind direction. The result from both modelswas in the form of a large ‘V’ open to the incoming wind direction. Inthis arrangement, a minimal number of turbines are directly in the wakesof other turbines and it is believed that, in potential flow terms,vortex sheets of turbines are created on either side of the V in orderto effectively channel the incoming wind. The two models predicteddifferent values for the improvement to performance obtained by thisarrangement over the performance obtained by the same number ofspatially isolated turbines. It is noted that the optimization programused for both programs is readily modified to allow optimization ofarrays with essentially arbitrary physical parameters.

The examples set forth above are provided to give those of ordinaryskill in the art a complete disclosure and description of how to makeand use the embodiments of the methods and systems for comparing windturbine arrays and providing configurations thereof of the disclosure,and are not intended to limit the scope of what the inventors regard astheir disclosure. Modifications of the above-described modes forcarrying out the disclosure can be used by persons of skill in the art,and are intended to be within the scope of the following claims.

Modifications of the above-described modes for carrying out the methodsand systems herein disclosed that are obvious to persons of skill in theart are intended to be within the scope of the following claims. Allpatents and publications mentioned in the specification are indicativeof the levels of skill of those skilled in the art to which thedisclosure pertains. All references cited in this disclosure areincorporated by reference to the same extent as if each reference hadbeen incorporated by reference in its entirety individually.

It is to be understood that the disclosure is not limited to particularmethods or systems, which can, of course, vary. It is also to beunderstood that the terminology used herein is for the purpose ofdescribing particular embodiments only, and is not intended to belimiting. As used in this specification and the appended claims, thesingular forms “a”, “an”, and “the” include plural referents unless thecontent clearly dictates otherwise. The term “plurality” includes two ormore referents unless the content clearly dictates otherwise. Unlessdefined otherwise, all technical and scientific terms used herein havethe same meaning as commonly understood by one of ordinary skill in theart to which the disclosure pertains.

A number of embodiments of the disclosure have been described.Nevertheless, it will be understood that various modifications can bemade without departing from the spirit and scope of the presentdisclosure. Accordingly, other embodiments are within the scope of thefollowing claims.

LIST OF REFERENCES

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What is claimed is:
 1. A V-shaped arrangement of turbines comprising atleast three pairs of turbines, wherein: the V-shaped arrangement isadapted to be oriented such that a prevalent crosswind enters theopening of the V-shaped arrangement and such that the prevalentcrosswind direction is substantially parallel to the axis bisecting theV-shape; each turbine in a pair of turbines is proximate to the otherturbine in the pair and non-proximate to other turbines in the array;the vertex of the V-Shaped arrangement comprises a clockwise rotatingturbine from which a first arm of the V-shaped arrangement extends in afirst direction, and a counterclockwise rotating turbine, proximate tothe clockwise rotating turbine, from which a second arm of the V-shapedarrangement extends; turbines comprised in the first arm extending inthe first direction are clockwise rotating turbines; and turbinescomprised in the second arm extending in the second direction arecounterclockwise rotating turbines.
 2. The V-shaped arrangement ofturbines according to claim 1, wherein an internal angle between thefirst arm and the second arm is between 30°-150°.
 3. The V-shapedarrangement of turbines according to claim 1, wherein an internal anglebetween the first arm and the second arm is approximately 90°.